Multiplicative Algorithm for computing
Optimum Designs
DAE Conference 2012
Roberto Dorta-Guerra, Raúl Martı́n Martı́n,
Mercedes Fernández-Guerrero and Licesio J. Rodrı́guez-Aragón
Optimum Experimental Design group
October 20, 2012
http://areaestadistica.uclm.es/oed
October 20th 2012 – 1 / 19
Overview
Optimum Design
Theory
Multiplicative
Algorithm
●
Optimum Design Theory.
●
Multiplicative Algorithm.
●
Multifactor Models:
Multifactor Models
Discriminating
between Models
✦ Enzyme Inhibition.
References
✦ pV T Measurements.
●
Discriminating between models.
✦ Adsorption Isoterms.
●
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Results Discussion.
October 20th 2012 – 2 / 19
Objectives
Optimum Design
Theory
●
Determination of optimum designs is not a
straightforward process, even for moderate examples.
●
New schemes for developing iterative algorithms, based
on the multiplicative algorithm
(Torsney and Martı́n-Martı́n, 2009), devoted to the
numerical construction of optimum designs are being
sought.
●
Provide experimenters with designs that would minimize
the experimental cost of measurements while obtaining
the most accurate characterization of the mechanisms.
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
References
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October 20th 2012 – 3 / 19
Optimum Design
Theory
❖ Optimum Design
Background
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
References
http://areaestadistica.uclm.es/oed
Optimum Design Theory
October 20th 2012 – 4 / 19
Optimum Design Background for nonlinear
models
Optimum Design
Theory
Consider the general non-linear regression model
❖ Optimum Design
Background
y = η(x; θ) + ε,
x ∈ X,
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
where the random variables ε are independent and normally
distributed with zero mean and constant variance σ 2 and θ is
the unknown parameter vector.
References
X , called design space.
Let Ξ be the set of probability distributions on the Borel sets
of X , then any ξ ∈ Ξ satisfying
Z
ξ(dx) = 1, ξ(xi ) ≥ 0, x ∈ X ,
X
is called a design measure, or an approximate design.
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October 20th 2012 – 5 / 19
Optimum Design Background for nonlinear
models
Optimum Design
Theory
❖ Optimum Design
Background
Multiplicative
Algorithm
The most common method for analyzing data from a
non-linear model is based on the use of the linear Taylor
series approximation of the response surface,
η(x; θ) ≈ η(x; θ0 ) + [∇η(x; θ)|θ0 ]t (θ − θ0 ),
Multifactor Models
Discriminating
between Models
References
where ∇ denotes the gradient with respect to θ and being θ0
a prior value of θ.
The variance-covariance matrix of the least square estimator
of θ is asymptotically approximated by the inverse of the
information matrix induced by ξ,
Z
M (ξ, θ) =
∇η(x; θ)[∇η(x; θ)]tdξ(x).
χ
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October 20th 2012 – 5 / 19
Optimum Design Background for nonlinear
models
Optimum Design
Theory
❖ Optimum Design
Background
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
References
Let φ be a real-valued function defined on the k × k symmetric
matrices and bounded above on {M (ξ, θ) : ξ ∈ Ξ}.
The optimum design problem is concerned with finding ξ ⋆
such that φ(M (ξ ⋆ , θ)) = minξ∈Ξ φ(M (ξ, θ)) which is called a
φ-optimum design.
Caratheodory’s theorem implies that for a model with k
parameters, a locally φ-optimum design supported at no
more than k(k + 1)/2 + 1 points exists.
For convenience, designs will be described using a two row
matrix,
x1 x2 . . . xN
ξ=
,
ξ 1 ξ 2 . . . ξN
being ξ1 , . . . , ξN non negative real numbers which sum up to
one.
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October 20th 2012 – 5 / 19
Optimum Design Background for nonlinear
models
Optimum Design
Theory
❖ Optimum Design
Background
The D−optimum criterion minimizes the volume of the
confidence ellipsoid of the parameters and is given by
φD [M (ξ, θ)] = det{M (ξ, θ)}−1/k ,
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
References
where k is the number of parameters in the model.
It is known that this criterion is a convex and non-increasing
function of the designs and so, designs with a small criterion
value are desirable.
A design that minimizes φD over all the designs on X is
called a D−optimum design.
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October 20th 2012 – 5 / 19
Optimum Design Background for nonlinear
models
Optimum Design
Theory
❖ Optimum Design
Background
Multiplicative
Algorithm
Since the criterion is convex, standard convex analysis
arguments using directional derivatives, when φ is
differentiable, show that a design ξ ⋆ is φ−optimum if and only
if
Multifactor Models
Discriminating
between Models
Fφ M (ξ ⋆ , θ), M (ξxj , θ) ≡ Fφ (ξ ⋆ , ej ) = dj −
References
N
X
ξj⋆ dj ≥ 0,
j=1
with equality at the support points of ξ ⋆ .
M (ξxj , θ) is the Information Matrix for a single observation at
the point xj , ej denotes the j-th unit vector in RN and
dj = ∂φ/∂ξxj .
The General Equivalence Theorem states that the variance
function evaluated using a φ-optimum design achieves its
maximum value at the support points of this design.
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October 20th 2012 – 5 / 19
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
References
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Multiplicative Algorithm
October 20th 2012 – 6 / 19
Numerical Aproximation of Optimum
Design
Optimum Design
Theory
Multiplicative
Algorithm
The purpose of experimental design is to select the design
points x1 , . . . , xN and the corresponding weights ξ1 , . . . , ξN so
that the design ξ is optimum for some criterion, φ.
Multifactor Models
Discriminating
between Models
References
A simple case is the determination of the best N -point
φ-optimum design. Let x1 , x2 , . . . , xN be its support points,
xi ∈ X = [a, b].
Let:
Wh =
xh − xh−1
, h = 1, . . . , N + 1,
b−a
where x0 = a and xN +1 = b. The variables Wh satisfy
Wh ≥ 0,
N
+1
X
Wh = 1.
h=1
Transformation was proposed by Torsney and Martı́n-Martı́n
(2009).
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October 20th 2012 – 7 / 19
Numerical Aproximation of Optimum
Design
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
References
Thus, we have an example of the following type of
optimization problem:
Minimize a criterion φ(W, ξ) over
P = W = (W1 , . . . , WN +1 ), ξ = (ξ1 , . . . , ξN ) :
Wh ≥ 0,
N
+1
X
Wh = 1 and ξj ≥ 0,
N
X
j=1
h=1
ξj = 1 .
Then the following simultaneous approaches are used
(r+1)
Wh
=
(r)
(r)
Wh g(Fh , δ1 )
,
PN +1 (r)
(r)
t=1 Wt g(Ft , δ1 )
(r)
(r+1)
ξj
(r)
ξj g(Fj , δ2 )
=P
N
(r)
(r)
i=1 ξi g(Fi , δ2 )
where r = 0, 1, . . . is the iteration number, g(Fh , δ1 ) and
g(Fj , δ2 ), are positive increasing, and if δ = 0, g(F, δ) are
constants.
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October 20th 2012 – 7 / 19
Numerical Aproximation of Optimum
Design
Optimum Design
Theory
Being Fj and Fh the vertex directional derivatives:
Multiplicative
Algorithm
Fh = Fφ (W, eh ) = dh −
Multifactor Models
Discriminating
between Models
Fj = Fφ (ξ, ej ) = dj −
References
X
X
Wh dh ,
ξj dj ,
∂φ
,
dh =
∂Wh
∂φ
.
dj =
∂ξj
Therefore if φ is a convex and differentiable function a design
ξ ⋆ will be optimum if and only if,
= 0, for Wh⋆ > 0
⋆
⋆
Fh = Fφ (W , eh ) =
≥ 0 for Wh⋆ = 0,
= 0, for ξ ⋆ (xj ) > 0
⋆
⋆
Fj = Fφ (ξ , ej ) =
≥ 0 for ξ ⋆ (xj ) = 0.
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October 20th 2012 – 7 / 19
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
Discriminating
between Models
Multifactor Models
References
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October 20th 2012 – 8 / 19
Multifactor Models
Optimum Design
Theory
Multiplicative
Algorithm
Searching for optimally designed experiments with more than
one independent factor is more complicated than for models
with a single factor.
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
●
Enzyme Inhibition.
●
Pressure, Volume and Temperature measurements.
Discriminating
between Models
References
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October 20th 2012 – 9 / 19
Enzyme Inhibition
Optimum Design
Theory
Multiplicative
Algorithm
Enzyme kinetics is the study of the chemical reactions that
are catalyzed by enzymes, with a focus on their reaction
rates.
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
Discriminating
between Models
References
The binding of an inhibitor can stop a substrate from entering
the enzyme’s active site and/or hinder the enzyme from
catalyzing its reaction. Inhibitor binding is either reversible or
irreversible.
Enzyme assays are laboratory procedures that measure the
rate of enzyme reactions.
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October 20th 2012 – 10 / 19
Enzyme Inhibition
Optimum Design
Theory
Multiplicative
Algorithm
This kinetic model is relevant to situations where very simple
kinetics can be assumed, (i.e. there is no intermediate or
product inhibition).
Multifactor Models
❖ Multifactor Models
Vmax [S]
V =
[I]
KM 1 + Ki + [S] 1 +
❖ Enzyme Inhibition
❖ pV T
Measurements
Discriminating
between Models
References
[I]
Ki′
●
Competitive inhibitors can bind to E, but not to ES.
●
Non-competitive inhibitors have identical affinities for E
and ES (Ki = Ki′ ).
●
Mixed-type inhibitors bind to both E and ES, but their
affinities for these two forms of the enzyme are different
(Ki 6= Ki′ ).
Unknown parameters θ = (Vmax , KM , Ki , Ki′ ).
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October 20th 2012 – 10 / 19
Enzyme Inhibition
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
Discriminating
between Models
Example: Considering Polifenol Oxidasa as enzyme,
4-Metylcatecol as substrate and Cianimic acid as a
competitive inhibitor, S × I = [5, 50] × [0, 1.2]µM and
θ0 = (V, Km , Ki ) = (131.4, 12.5, 0.42).
(8.33, 0) (48, 26, 1.2) (50, 0)
⋆
ξD =
.
0.33
0.33
0.33
References
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October 20th 2012 – 10 / 19
Enzyme Inhibition
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
D-optima designs for these type of models have been
analytically computed by Bogacka et al. (2011).
Comparison of convergence rate of the Multiplicative
Algorithm (red) with Wynn-Fedorov Algorithm (blue) have
been obtained:
Discriminating
between Models
References
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October 20th 2012 – 10 / 19
pV T Measurements
Optimum Design
Theory
Multiplicative
Algorithm
The characterization of volume or density as a function of
temperature and pressure is particularly important for the
design of industrial plants, pipelines and pumps.
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
In order to correlate the density values over the temperature
and pressure intervals, the following Tait-like equation is
used,
Discriminating
between Models
References
E(ρ) = η(p, T ; θ) =
ρ0 (T )
B(T )+p
1 − C(T ) log B(T
)+p0
,
var(ρ) = σ 2 .
ρ0 (T ) is either a linear function of the required degree or a
non-linear function, known as Rackett equation.B(T ) and
C(T ) are linear functions.
X = P × T , where P and T are permissible ranges of values
for p and T . Being θt = (A0 , . . . , B0 , . . . , C0 , . . . ) the set of
unknown parameters.
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October 20th 2012 – 11 / 19
pV T Measurements
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
Example: In order to characterize changes of density of
1-phenylundecane.
X = P × T = [0.1, 65]M P a × [293.15, 353, 15]K and the set of
best guesses of the parameters are obtained from
Milhet et al. (2005), θ0 = (A0 , A1 , A2 , A3 , B0 , B1 , B2 , C)t .
❖ pV T
Measurements
Discriminating
between Models
References
8
7
6
5
0
340
20
Pressure
320 Temperature
40
60
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300
October 20th 2012 – 11 / 19
pV T Measurements
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
Discriminating
between Models
References
⋆ is supported at 11 points with
The obtained design, ξD
different proportions of observations for each point. In this
case, 8 of the support points lie at the boundaries of the
design space X , while two of them belong to its interior.
t


(0.1, 293.15)
0.12 







(0.1, 311.99)
0.07 








(0.1,
333.55)
0.10








(0.1,
353.15)
0.12








 (28.58,308.91) 0.05 
⋆
.
(28.58,339.50) 0.07
ξD
=




(28.58,
353.15)
0.07









(65, 293.15)
0.12 








(65,
311.61)
0.07








(65,
334.53)
0.09




 (65, 353.15)
0.12 
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October 20th 2012 – 11 / 19
pV T Measurements
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
Discriminating
between Models
Example:In the work of Outcalt and Laesecke (2010),
measurements are taken over JP-10. Because of its high
thermal stability, high energy density, low cost, and
widespread availability, JP-10 is being investigated as a fuel
to be used in pulse-detonation engines.
In this case the dependence of ρ0 is characterized by the
Racket equation:
References
ρ0 (T ) =
[1+(1−(T /CR ))DR ]
AR /BR
.
Measurements are taken over
X = P × T = [0.083, 30]M P a × [270, 470]K and the set of
best guesses are obtained from Outcalt and Laesecke
(2010), θ0 = (AR , BR , CR , DR , B0 , B1 , B2 , C)t .
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October 20th 2012 – 11 / 19
pV T Measurements
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
Discriminating
between Models
References
⋆ is supported at 10 points with
The obtained design, ξD
different proportions of observations for each point. 9 of the
support points lie in the boundary while only one is in the
interior of X .

t
(0.083, 270)
0.11 







(0.083,
334.95)
0.08








(0.083,
421.72)
0.12








(0.083,
470)
0.12






(12.51,439.1)
0.02
⋆
.
ξD
=
(12.60, 470)
0.12 









(30,
270)
0.11








(30,
337.94)
0.09







(30, 425.91)
0.11 




(30, 470)
0.12
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October 20th 2012 – 11 / 19
pV T Measurements
Optimum Design
Theory
Multiplicative
Algorithm
The Equivalence Theorem proves D−optimality of the
design. Figure shows the attainment of equality to k = 8 at
the support points.
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
Discriminating
between Models
8
7
References
6
5
4
0
450
10
400
Pressure
350
20
Temperature
300
30
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October 20th 2012 – 11 / 19
pV T Measurements
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
Discriminating
between Models
References
For the extension of the multiplicative algorithm to a
two-factor non-linear model, the transformation of pressure
values, pi , i = 1, . . . , N , into a marginal distribution or set of
weights Wph , h = 1, . . . , N + 1, is proposed.
Then, the corresponding temperatures T paired with each
value pi , Ts|pi , s = 1, . . . , q, are transformed into a conditional
distribution WTs |pi , s = 1, . . . , q.
On the other hand, design weights need to be
simultaneously determined. These conditional and marginal
distributions must be optimally chosen.
According to Caratheodory’s theorem we initially assume
k(k + 1)/2 + 1 support points. The optimization problem can
be stated as a problem with respect to N + 2 distributions.
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October 20th 2012 – 11 / 19
pV T Measurements
Optimum Design
Theory
Tb
WTq+1 pi
Multiplicative
Algorithm
Hpi ,Tq pi L
Multifactor Models
❖ Multifactor Models
❖ Enzyme Inhibition
Discriminating
between Models
.
.
.
T=@Ta ,Tb D
❖ pV T
Measurements
.
.
.
Hpi ,T2 pi L
References
W T 2 pi
Hpi ,T1 pi L
W T 1 pi
W p3
W p1 W p2
WpN+1
Ta
pa
p1
p2
p3
...
pi
...
pN
pb
P=@ pa , pb D
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October 20th 2012 – 11 / 19
pV T Measurements
Optimum Design
Theory
Multiplicative
Algorithm
This leads to the following simultaneous multiplicative
iterations,
Multifactor Models
(r)
❖ Multifactor Models
❖ Enzyme Inhibition
❖ pV T
Measurements
Discriminating
between Models
References
(r)
Wph g(Fph , δ1 )
Wp(r+1)
=P
h
N +1
(r+1)
WTs |pi =
(r+1)
ξt
, h = 1, . . . , N + 1,
(r)
(r)
W
g(F
pj
pj , δ1 )
j=1
(r)
(r)
WTs |pi g(FTs |pi , δ2 )
,
Pq+1 (r)
(r)
l=1 WTl |pi g(FTl |pi , δ2 )
(r)
(r)
ξt g(Ft , δ3 )
= PqN
s = 1, . . . , q + 1, i = 1, . . . , N,
(r)
(r)
ξ
g(F
i=1 i
i , δ3 )
, t = 1, . . . , qN,
where g(F, δ) = Φ(δF ), and Fp , FT |p , Ft are the vertex
directional derivatives.
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October 20th 2012 – 11 / 19
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
Discriminating between Models
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
References
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October 20th 2012 – 12 / 19
Discriminating between models
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
Atkinson and Fedorov (1975a,b) introduced the so called
T-optimality criterion which has an interesting statistical
interpretation as the power of a test for the fit of a second
model when the other is considered as the true model.
Usually there is no closed form for the T-optimum design and
it must be computed through an iterative procedure.
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
References
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October 20th 2012 – 13 / 19
Discriminating between models
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
References
Let us assume that the function η(xi , θ) coincides with either
η1 (x, θ1 ) or η2 (x, θ2 ) partially known functions where
θ1 ∈ Ω1 ⊂ Rm1 and θ2 ∈ Ω2 ⊂ Rm2 are the unknown
parameter vectors.
Let us assume that η(x, θ) = η1 (x, θ1 ) is the true model of the
process with parameters θ1 known and η2 is the rival model.
Atkinson and Fedorov (1975a,b) introduced the notion of
T -optimality.
X
T21 (ξ) = min
[η(xi , θ) − η2 (xi , θ2 )]2 ξ(xi ).
θ2 ∈Ω2
xi ∈χ
The design ξ ∗ which maximizes T21 (ξ) is called the T -optimal
design.
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October 20th 2012 – 13 / 19
Discriminating between models
Optimum Design
Theory
A design for which the optimization problem
Multiplicative
Algorithm
θb2 ≡ arg min
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
References
θ2 ∈Ω2
k
X
2
η(xi ) − η2 (xi , θ2 ) ξi
i=1
has no unique solution is a singular design, otherwise is
called regular.
For regular designs the Equivalence Theorem is applicable
with the implication: a design ξ ∗ is T -optimal if and only if,
k
2 X
2
b
Fj (ξ) = η(xj , θ)−η2 (xj , θ2 ) −
η(xi , θ)−η2 (xi , θ2 ) ξi ≤ 0,
i=1
with equality at the support points.
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October 20th 2012 – 13 / 19
Atkinson-Fedorov Algorith
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
Based on the Equivalence Theorem Atkinson and Fedorov
(1975a,b) provided the following algorithm:
x1 . . . xk
(0)
● For a given initial design ξk =
ξ 1 . . . ξk
determine
k
X
2
b
η(xi ) − η2 (xi , θ2 ) ξi
θ2 = arg min
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
θ2 ∈Ω2
●
i=1
Find the point
References
xk+1
http://areaestadistica.uclm.es/oed
= arg max η(x) − η2 (x, θb2 )
x∈χ
2
October 20th 2012 – 14 / 19
Atkinson-Fedorov Algorith
Optimum Design
Theory
●
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
Let ξxk+1 be a design with measure concentrated at the
single point xk+1 ,
xk+1
ξxk+1 =
1
A new design is constructed in the following way:
❖ Multiplicative
Algorithm
ξk+1 = (1 − αk+1 )ξk + αk+1 ξxk+1
❖ Adsorption
Isoterms
❖ Results
Discussion
References
where typical conditions
for the sequence
P∞
P∞ 2 {αk } are
limk→∞ αk = 0, k=0 αk = ∞, k=0 αk < ∞.
In our work αs = 1/(s + 1) has been used.
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October 20th 2012 – 14 / 19
Multiplicative Algorithm
Optimum Design
Theory
Multiplicative
Algorithm
Assuming that each ξ is regular, i.e. Ω2 (ξ) = θb2 , this first
optimization problem was solved using a quasi-Newton
algorithm through the FORTRAN IMLS routine DBCONF.
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
References
Then we wish to choose a design ξ optimally, that is, we wish
to determine both the support points and the design weights
optimally
X
⋆
ξ = arg max
[η(xi , θ)−η2 (xi , θb2 )]2 ξ(xi ) = arg max Tc
21 (ξ).
(x,ξ(x))
xi ∈χ
(x,ξ(x))
In this case, the equivalence theorem says nothing about the
number of support points of an optimal design. We consider
designs with L support points, x1 , . . . , xL , where L is an
appropriate number for each problem.
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October 20th 2012 – 15 / 19
Multiplicative Algorithm
Optimum Design
Theory
Let
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
xt − xt−1
Wt =
t = 1, . . . , L + 1
b−a
where x0 = a and xL+1 = b. We have transformed from L
variables
P to L + 1 variables, but these must satisfy Wt ≥ 0
and t Wt = 1.
As in Torsney and Martı́n-Martı́n (2009) we have an
optimization problem with respect to two distributions one
defined by the design points and one defined by the design
weights.
References
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October 20th 2012 – 15 / 19
Multiplicative Algorithm
Optimum Design
Theory
Multiplicative
Algorithm
The multiplicative algorithm extends naturally to the two
simultaneous multiplicative iterations:
Multifactor Models
(r+1)
Wt
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
References
(r)
=P
L+1
(r)
(r+1)
ξj
(r)
h=1 Wh g(Fh , δ1 )
(r)
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
(r)
Wt g(Ft , δ1 )
(r)
ξj g(Fj , δ2 )
=P
L
(r)
(r)
i=1 ξi g(Fi , δ2 )
Being Fj and Fh the vertex directional derivatives of T21 at W
and ξ :
X
Fh = FT21 (W, eh ) = dh −
Wh dh
X
Fj = FT21 (ξ, ej ) = dj −
ξj dj
Where dh =
http://areaestadistica.uclm.es/oed
∂T21
∂Wh
and dj =
∂T21
∂ξj .
October 20th 2012 – 15 / 19
Multiplicative Algorithm
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
The same first order conditions for a local maximum are used:
= 0, for Wh∗ > 0
∗
∗
Fh = FT21 (Wh , eh ) =
≥ 0, for Wh∗ = 0
= 0, for ξj∗ > 0
∗
∗
Fj = FT21 (ξ , ej ) =
≥ 0, for ξj∗ = 0
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
References
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October 20th 2012 – 15 / 19
Adsorption Isoterms
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
References
The modeling of the adsorption phenomena in many chemical
and industrial processes has proved to be of great interest.
The two models used in the literature to describe the
relationship between the amount of gas or water adsorbed,
we , in terms of water activity aw for multilayer adsorption
phenomena are the Brunauer-Emmett-Teller (BET) model
and the extension known as Guggenheim-Anderson-de Boer
(GAB) model.
BET model: E[we ] =
t
unknown parameters: θB
=
GAB model: E[we ] =
t
unknown parameters: θG
=
http://areaestadistica.uclm.es/oed
wmB cB aw
,
(1 − aw )(1 + (cB − 1)aw )
(wmB , cB ).
wmG cG kaw
(1 − kaw )(1 + (cG − 1)kaw )
(wmG , cG , k).
October 20th 2012 – 16 / 19
Adsorption Isoterms
Optimum Design
Theory
Multiplicative
Algorithm
Cepeda et al. (1999) studied water sorption behavior of
coffee for predicting hygroscopic properties as well as
designing units for its optimum preservation, storage, etc.
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
The results at 25 C0 for the GAB adsorption isotherm were
wmG = 0.03445 g of H2 O adsorbed/g of coffee, cG = 11.70
and k = 0.994.
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
References
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October 20th 2012 – 16 / 19
Adsorption Isoterms
Optimum Design
Theory
Multiplicative
Algorithm
Rodrı́guez-Aragón and López-Fidalgo (2007) provided the
T-optimal design considering the GAB model as the true
model using the Atkinson-Fedorov algorithm.
Multifactor Models
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
❖ Multiplicative
Algorithm
❖ Adsorption
Isoterms
❖ Results
Discussion
References
After 182 iterations of the algorithm, a design supported at
three experimental points was obtained, with a lower
efficiency bound of 0.998.
0.056 0.62 0.8
ξ∗ =
0.150 0.57 0.28
For the same problem the new approach was applied. After
129 iterations of this algorithm, the T-optimum design was
obtained, with a lower efficiency bound of 0.998.
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October 20th 2012 – 16 / 19
Results Discussion
Optimum Design
Theory
●
Multiplicative
Algorithm
Multifactor Models
Advantages of computing optima designs with the
Multiplicative Algorithm:
✦ Effectiveness to compute D−optimum designs for
Discriminating
between Models
❖ Discriminating
between models
❖ Atkinson-Fedorov
Algorith
multifactor modesl.
✦ T −optimization with simpler computations.
❖ Multiplicative
Algorithm
●
❖ Adsorption
Isoterms
❖ Results
Discussion
Locally optimum designs: designs depend on the initial
best guesses of the parameters.
●
Open issues: Choice of function g and constants δ,
Torsney and Mandal (2006).
●
Possibility of computing θ̂2 using the multiplicative
algorithm.
References
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October 20th 2012 – 17 / 19
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
References
http://areaestadistica.uclm.es/oed
References
October 20th 2012 – 18 / 19
References
Optimum Design
Theory
Multiplicative
Algorithm
Multifactor Models
Discriminating
between Models
References
• B. Torsney and R. Martı́n-Martı́n. Multiplicative algorithms for computing optimum
designs. Journal of Statistical Planning and Inference, 139(12):3947–3961, 2009
• B. Bogacka, M. Patan, P. Johnson, K. Youdim, and A. C. Atkinson. Optimum
designs for enzyme inhibition kinetic models. Journal of Biopharmaceutical
Statistics, 21:555–572, 2011
• M. Milhet, A. Baylaucq, and C. Boned. Volumetric properties of 1-phenyldecane
and 1-phenylundecane at pressures to 65 MPa and temperature between 293.15
and 353.15 K. J. Chem. Eng. Data, 50(4):1430–1433, 2005
• S. L. Outcalt and A. Laesecke. Measurement of density and speed of sound of
JP-10 and comparsion to rocket propellants and jet fuels. Energy Fuels, 25:
1132–1139, 2010
• A. C. Atkinson and V. V. Fedorov. The design of experiments for discriminating
between two rival models. Biometrika, 62:57–70, 1975a
• A. C. Atkinson and V. V. Fedorov. Optimal design: Experiments for discriminating
between several models. Biometrika, 62:289–303, 1975b
• E. Cepeda, R. Ortı́z de Latierro, M. J. San José, and M. Olazar. Water sorption
isotherms of roasted coffee and coffee roasted with sugar. Int. J. Food Sci. Technol.,
34, 1999
• L. J. Rodrı́guez-Aragón and J. López-Fidalgo. T −, D− and c− optimum designs
for BET and GAB adsorption isotherms. Chemom. Intell. Lab. Syst., 89:36–44, 2007
• B. Torsney and S. Mandal. Two classes of multiplicative algorithms for
construction optimizing distributions. Computational Statistics and Data Analysis,
51:1591–1601, 2006
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October 20th 2012 – 19 / 19