Optimal Designs for a Modified Exponential Model Juan M. Rodr´ ıguez-D´ ıaz Mar´ ıa Teresa Santos-Mart´ ın 1 Historical notes From the middle of the 19th century on several equations relating the rate constant k of a chemical reaction to the temperature T have appeared in literature. All of them were developed experimentally and the most popular ones try to fit linearly ln ( k ) against T , 1 /T or ln ( T ). The fact that this different plots gave reasonably good linear fits with the same data was due to the narrow temperature ranges usually employed in kinetic studies (see [Laidler (1984)]), but after the controversy on the models maintained during several decades all these equations were gradually dropped out but two of them: the Arrhenius and Modified-Arrhenius ones. The main reason for the rejection of the remaining equations was the lack of theoretical justification for them, while the Arrhenius ones were well explained theoretically. 1.1 The Arrhenius equation The Arrhenius equation is widely accepted as the right tool to describe the influence of temperature on the rates of chemical processes. It was first used by Svante Arrhenius in 1884 in his studies of dissociation of electrolytes, but later on applied to describe the relationship between temperature and the rates of chemical reactions and many other physical processes such as diffusion, thermal and electrical conductivity, viscosity, etc. The integrated form of the equation is ln( k ) = A ′ − β T , where β = E/R , with E the activation energy and R the gas constant . By taking expo- nentials and making the change T = 1 /X it comes to be the exponential model E [ k ] = Ae − βx + ε (1) where A = e A ′ > 0 is the frequency factor , and β > 0. The optimal designs for this model have been studied in [Han and Chaloner (2003)] for independent and normally-distributed errors with constant variance and in [Rodr´ ıguez-Torreblanca and Rodr´ ıguez-D´ ıaz (2007)] for different variance structures. Also optimal and compound designs specifically for the Arrhenius equation as well as a study of the efficiency of some designs used in literature can be found in [Rodr´ ıguez-Arag´ on and L´ opez-Fidalgo (2005)]. However, for the analysis of more precise rate-temperature data, particularly in studies covering a wide temperature range, it is usual to allow A ′ to be temperature-dependent, proportional to ln ( T ), or equivalently A proportional to X raised to a power m producing E [ k ] = ax m e − βx + ε (2) where a > 0 is now temperature independent and β > 0, the so called Modified-Arrhenius (MA) model. Nowadays (see [Laidler (1984)]), the procedure often employed is to use 1
model (1) for data of lower precision or where the temperature range is limited, and to analyze more precise data by using model (2). This will be the model we will work with in this paper. 1.2 Optimal Design theory In Optimal Experimental Design theory the work with non-linear models like the MA one is far more complex than the case of linear ones, due to the fact that in the first case the best design will depend on the values of the unknown parameters. A non-linear model can be written as Y = η ( x, θ ) + ǫ, (3) where Y denotes the observation, x ∈ X = ( x min , x max ) is the independent variable, θ = ( θ 1 , θ 2 , . . . θ k ) ∈ Θ is the vector of unknown parameters, and η ( x, θ ) represents a function that is non-linear respect to θ . The ǫ term stands for the random errors, coming from either the experiment itself or due to a bad choice of the model. In the homoscedastic case the errors are assumed to be normally distributed, with zero mean and constant variance. The heteroscedastic model allows a more complex structure for the variance, and the treatment is more difficult. A design is a collection of points of the independent variable, { x 1 , x 2 , . . . , x N } , where N is the size of the design. It can be written taking the n different points (called the support points) and for each one the proportion (weight) than it has in the design, that is � x 1 x 2 . . . n � x n � ξ n = , p i = 1 , x i ∈ X , i = 1 , . . . , n p 1 p 2 . . . p n i =1 where n i = Np i is the frecuency of point x i in the design. From this point of view, an approximate design can be defined as any probability measure in X with finite support. Let p ( Y | x, θ ) be the probability density function for the random variable Y with model (3). The ( i, j ) element of the Fisher information matrix for a given x is, � − ∂ 2 log p ( y | x, θ ) � I ( x, θ ) = E . ∂θ i ∂θ j For n independent samples in x = x 1 , . . . , x n , the corresponding matrix is n � I ( x, θ ) = I ( x i , θ ) . i =1 The information matrix for an approximate design ξ is M ( ξ, θ ) = E ( I (˜ x, θ )), where ˜ x is a random vector with probability distribution function ξ . The information matrix becomes the main tool when looking for the optimal design for the experiment. When the function η ( x, θ ) in model (3) is differentiable with continuous derivative for every parameter θ i , the information matrix for a design ξ will be n � p i ∇ η ( x i , θ ) ∇ η ( x i , θ ) T , M ( ξ, θ ) = i =1 2
where � ∂η ( x, θ ) � , ∂η ( x, θ ) , . . . , ∂η ( x, θ ) ∇ η ( x, θ ) = , ∂θ 1 ∂θ 2 ∂θ k is the gradient vector of η ( x, θ ). The last expression is for the homoscedastic case. When a heteroscedastic model is assumed, with error variance given by var ( Y | x ) = σ 2 /ω ( x ), being ω ( x ) a non-negative function called the efficiency function , the information matrix is n � p i ω ( x i ) ∇ η ( x i , θ ) ∇ η ( x i , θ ) T . M ( ξ, θ ) = (4) i =1 In both cases, when dealing with non-linear models the information matrix depends on the unknown parameters. To solve this problem some kind of additional information is needed, either an initial value for the parameters or a priori distribution for them. In any case, the optimal design will be a function of these initial values or distributions. The inverse of the information matrix is proportional to the covariance matrix of the estimators of the parameters of the model. Optimal experimental designs typically minimize some convex function of the inverse of the information matrix. If the aim is to estimate the parameters in the model, a reasonable criterion is D -optimality, that focuses in the determinant of the information matrix. A design ξ is D-optimal if maximices this determinant, what is equivalent to minimize the one of the covariance matrix (the generalized variance of the parameter estimators). A D -optimal design minimizes the volume of the confidence ellipsoid of the parameters. The General Equivalence Theorem is a useful tool for checking that a design is optimal. In the case of D -optimality, it can be stated as follows: If k is the number of parameters of the model and θ = θ 0 , then the design ξ ∗ is D-optimal if and only if ω ( x ) ∇ η ( x, θ 0 ) T M − 1 ( ξ ∗ , θ 0 ) ∇ η ( x, θ 0 ) ≤ k x ∈ X and the inequality becomes equality in the support points of ξ ∗ . If we are interested in looking for the best estimator for a linear combination of para- meters then c -optimization is the right criterion function. It is specially used for vectors (1,0,...,0),...,(0,...,0,1), getting optimal designs for the estimation of each parameter. For c ∈ R k , a design ξ is c-optimal if maximices − c T M ( ξ, θ ) − 1 c what is equivalent to minimize V ar ( c T ˆ θ ). By the General Equivalence Theorem for c -optimality, the design ξ ∗ is c -optimal if and only if ( ∇ η ( x, θ 0 ) T M − 1 ( ξ ∗ , θ 0 ) c ) 2 ≤ c T M − 1 ( ξ ∗ , θ 0 ) c ∀ x ∈ X However, c-optimal designs can be constructed geometrically using Elfving’s Theorem [Elfving (1952)]. The exponential regression model is one of the most used in literature since models the behaviour of laboratory cultives, the time-life of dispositives, etc. The exponential distri- bution describes the time for a continuous process to change state. Optimal experimen- tal designs for exponential regression models have been studied in [Box and Lucas (1953)], [Melas (1978)], [Dette and Sperlich (1994)], [Mukhopadhyay and Haines (1995)], [Dette and Neugebauer [Dette and Neugabauer (1997)], [Han and Chaloner (2003)] and [Rodr´ ıguez-Torreblanca and Ro 3
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