Experimental design for four mixture components with process variables

The experimental design to model the response of a mixture in four components in the presence of process variables is considered. Two different blocks of blends that are orthogonal for linear or quadratic blending are D-optimized. The two orthogonal blocks of blends are generalized and D-optimized in some cases (and possibly Doptimized in others) to deal with restrictions on the blending component proportions. The pair of orthogonal D-optimal blocks of blends can be used with an arbitrary number of process variables, and requires a reduced number of observations.     

Mixture experiments with process variables: d-optimal orthogonal experimental designs

Blending experiments with mixture in the presence of process variables are considered. We present an experimental design for quadratic (or linear) blending. The design in two orthogonal blocks is D-optimized in the case where there are no restrictions on the blending in two orthogonal blocks is presented when there are arbitrary restrictions on the blending components. The pair of orthogonal blocks can be used with and arbitrary number of process variables. The number of design points needed when different orthogonal blocks are used is usually smaller than when a single block is repeated at the various process variables levels.     

A characterization of a universally optimal design within a class of block designs

It is shown that within the class of connected binary designs with arbitrary block sizes and arbitrary replications only a symmetic balanced incomplete block design produces a completely symmetric information matrix for the treatment effects whenever the number of blocks is equal to the number of treatments and the number of experimental units is an integer multiple of the number of treatments. Such a design is known to be universally optimal.     

Efficient arrangements of two-level orthogonal arrays in two and four blocks

When orthogonal arrays are used in practical applications, it is often difficult to perform all the designed runs of the experiment under homogeneous conditions. The arrangement of factorial runs into blocks is usually an action taken to overcome such obstacles. However, an arbitrary configuration might lead to spurious analysis results. In this work, the nice properties of two-level orthogonal arrays are taken into consideration and an effective method for arranging experimental runs into two and four blocks of the same size is proposed. This method is based on the so-called J-characteristics of the corresponding array. General theoretical results are given for studying up to four experimental factors in two blocks, as well as for studying up to three experimental factors in four blocks. Finally, we provide best blocking arrangements when the number of the factors of interest is larger, by exploiting the known lists of non-isomorphic orthogonal arrays with two levels and various run sizes.     

Projection designs for mixture experiments in orthogonal blocks

Mixture experiments are often carried out in the presence of process variables, such as days of the week or different machines in a manufacturing process, or different ovens in bread and cake making. In such experiments it is particularly useful to be able to arrange the design in orthogonal blocks, so that the model in tue mixture vanauies may ue iitteu inucpenuentiy or tne UIOCK enects mtrouuceu to take account of the changes in the process variables. It is possible in some situations that some of the ingredients in the mixture, such as additives or flavourings, are present in soian quantities, pernaps as iuw a.s 5% ur even !%, resulting in the design space being restricted to only part of the mixture simplex. Hau and Box (1990) discussed the construction of experimental designs for situations where constraints are placed on the design variables. They considered projecting standard response surface designs, including factorial designs and central composite designs, into the restricted design space, and showed that the desirable property of block orthogonality is preserved by the projections considered. Here we present a number of examples of projection designs and illustrate their use when some of the ingredients are restricted to small values, such that the design space is restricted to a sub-region within the usual simplex in the mixture variables.     

Blocking in two-level non-regular fractional factorial designs

In this paper we consider screening experiments where a two-level fractional factorial design is to be used to identify significant factors in an experimental process and where the runs in the experiment are to occur in blocks of equal size. A simple method based on the foldover technique is given for constructing resolution IV orthogonal and non-orthogonal blocked designs and examples are given to illustrate the process.     

Interpretable dimension reduction

The analysis of high-dimensional data often begins with the identification of lower dimensional subspaces. Principal component analysis is a dimension reduction technique that identifies linear combinations of variables along which most variation occurs or which best “reconstruct” the original variables. For example, many temperature readings may be taken in a production process when in fact there are just a few underlying variables driving the process. A problem with principal components is that the linear combinations can seem quite arbitrary. To make them more interpretable, we introduce two classes of constraints. In the first, coefficients are constrained to equal a small number of values (homogeneity constraint). The second constraint attempts to set as many coefficients to zero as possible (sparsity constraint). The resultant interpretable directions are either calculated to be close to the original principal component directions, or calculated in a stepwise manner that may make the components more orthogonal. A small dataset on characteristics of cars is used to introduce the techniques. A more substantial data mining application is also given, illustrating the ability of the procedure to scale to a very large number of variables.     

Connectedness of complete block designs under an interference model

We consider an experiment with fixed number of blocks, in which a response to a treatment can be affected by treatments from neighboring units. For such experiment the interference model with neighbor effects is studied. Under this model we study connectedness of binary complete block designs. Assuming the circular interference model with left-neighbor effects we give the condition for minimal number of blocks necessary to obtain connected design. For a specified class of binary, complete block designs, we show that all designs are connected. Further we present the sufficient and necessary conditions of connectedness of designs with arbitrary, fixed number of blocks.     

Optimal orthogonal block designs for a quadratic mixture model for three components

In experiments with mixtures that involve process variables, if the response function is expressed as the sum of a function of mixture components and a function of process variables, then the parameters in the mixture part and in the process part can be estimated independently using orthogonal block designs. This paper is concerned with such a block design for parameter estimation in the mixture part of a quadratic mixture model for three mixture components. The behaviour of the eigenvalues of the moment matrix of the design is investigated in detail, the design is optimized according to E- and Aoptimality criteria, and the results are compared together with a known result on Doptimality. It is found that this block design is robust with respect to these diff erent optimality criteria against the shifting of experimental points. As a result, we recommend experimental points of the form (a, b, c) in the simplex S2, where c=0, b=1-a, and a can be any value in the range 0.17+/-0.02.     

Nearly optimal orthogonally blocked desings for a quadratic mixture model with q components

In experiments with mixtures involving process variables, orthogonal block designs may be used to allow estimation of the parameters of the mixture components independently of estimation of the parameters of the process variables. In the class of orthogonally blocked designs based on pairs of suitably chosen Latin squares, the optimal designs consist primarily of binary blends of the mixture components, regardless of how many ingredients are available for the mixture. This paper considers ways of modifying these optimal designs so that some or all of the runs used in the experiment include a minimum proportion of each mixture ingredient. The designs considered are nearly optimal in the sense that the experimental points are chosen to follow ridges of maxima in the optimality criteria. Specific designs are discussed for mixtures involving three and four components and distinctions are identified for different designs with the same optimality properties. The ideas presented for these specific designs are readily extended to mixtures with q>4 components.     

Design and analysis of mixture experiments with process variable

A mixture experiment is an experiment in which the response is assumed to depend on the relative proportions of the ingredients present in the mixture and not on the total amount of the mixture. In such experiment process, variables do not form any portion of the mixture but the levels changed could affect the blending properties of the ingredients. Sometimes, the mixture experiments are costly and the experiments are to be conducted in less number of runs. Here, a general method for construction of efficient mixture experiments in a minimum number of runs by the method for projection of efficient response surface design onto the constrained region is obtained. The efficient designs with a less number of runs have been constructed for 3rd, 4th, and 5th component of mixture experiments with one process variable.     

Optimal regular graph designs

A typical problem in optimal design theory is finding an experimental design that is optimal with respect to some criteria in a class of designs. The most popular criteria include the A- and D-criteria. Regular graph designs occur in many optimality results, and if the number of blocks is large enough, an A-optimal (or D-optimal) design is among them (if any exist). To explore the landscape of designs with a large number of blocks, we introduce extensions of regular graph designs. These are constructed by adding the blocks of a balanced incomplete block design repeatedly to the original design. We present the results of an exact computer search for the best regular graph designs and the best extended regular graph designs with up to 20 treatments v, block size \(k \le 10\) and replication r \(\le 10\) and \(r(k-1)-(v-1)\lfloor r(k-1)/(v-1)\rfloor \le 9\).     

Mixture designs for five components in orthogonal blocks

The methods developed by John and Draper et al. of partitioning the blends (runs) of four mixture components into two or more orthogonal blocks when fitting quadratic models are extended to mixtures of five components. The characteristics of Latin squares of side five are used to derive rules for reliably and quickly obtaining designs with specific properties. The designs also produce orthogonal blocks when higher order models are fitted.     

Partially A-optimal blocked multi-factor designs

In this paper, we propose a partially A-optimal criterion for block designs where multiple factors are arranged. The number of levels of each factor is assumed to be arbitrary and unequal block sizes are allowed. A sufficient condition is derived for a design to be partially A-optimal among all feasible designs. Then the properties of the selected design and its relation with orthogonal arrays are studied. Methods of constructing designs satisfying the sufficient condition are also given.     

Orthogonal blocking of response surface split-plot designs

When all experimental runs cannot be performed under homogeneous conditions, blocking can be used to increase the power for testing the treatment effects. Orthogonal blocking provides the same estimator of the polynomial effects as the one that would be obtained by ignoring the blocks. In many real-life design scenarios, there is at least one factor that is hard to change, leading to a split-plot structure. This paper shows that for a balanced ordinary least square–generalized least square equivalent split-plot design, orthogonal blocking can be achieved. Orthogonally blocked split-plot central composite designs are constructed and a catalog is provided.     

Balanced incomplete Latin square designs

Latin squares have been widely used to design an experiment where the blocking factors and treatment factors have the same number of levels. For some experiments, the size of blocks may be less than the number of treatments. Since not all the treatments can be compared within each block, a new class of designs called balanced incomplete Latin squares (BILS) is proposed. A general method for constructing BILS is proposed by an intelligent selection of certain cells from a complete Latin square via orthogonal Latin squares. The optimality of the proposed BILS designs is investigated. It is shown that the proposed transversal BILS designs are asymptotically optimal for all the row, column and treatment effects. The relative efficiencies of a delete-one-transversal BILS design with respect to the optimal designs for both cases are also derived; it is shown to be close to 100%, as the order becomes large.     

A class of single replicate orthogonal designs

A simple method is given to calculate the number of degrees of freedom confounded with blocks of a specific factorial effect in a single replicate orthogonal design. Two classes of designs having partial orthogonality are also discussed     

Effect of missing plots in some response surface designs

The effect of one or more missing observations for response surface designs arranged in blocks are examined in this paper. The resu lts as applied to a central composite design with orthogonal blocking, and an equirdial design with orthogonal blocking, are reported.     

Mixture designs for constrained components in orthogonal blocks

It is often the case in mixture experiments that some of the ingredients, such as additives or flavourings, are included with proportions constrained to lie in a restricted interval, while the majority of the mixture is made up of a particular ingredient used as a filler. The experimental region in such cases is restricted to a parallelepiped in or near one corner of the full simplex region. In this paper, orthogonally blocked designs with two experimental blends on each edge of the constrained region are considered for mixture experiments with three and four ingredients. The optimal symmetric orthogonally blocked designs within this class are determined and it is shown that even better designs are obtained for the asymmetric situation, in which some experimental blends are taken at the vertices of the experimental region. Some examples are given to show how these ideas may be extended to identify good designs in three and four blocks. Finally, an example is included to illustrate how to overcome the problems of collinearity that sometimes occur when fitting quadratic models to experimental data from mixture experiments in which some of the ingredient proportions are restricted to small values.     

Batch sequential designs for computer experiments

Computer models simulating a physical process are used in many areas of science. Due to the complex nature of these codes it is often necessary to approximate the code, which is typically done using a Gaussian process. In many situations the number of code runs available to build the Gaussian process approximation is limited. When the initial design is small or the underlying response surface is complicated this can lead to poor approximations of the code output. In order to improve the fit of the model, sequential design strategies must be employed. In this paper we introduce two simple distance based metrics that can be used to augment an initial design in a batch sequential manner. In addition we propose a sequential updating strategy to an orthogonal array based Latin hypercube sample. We show via various real and simulated examples that the distance metrics and the extension of the orthogonal array based Latin hypercubes work well in practice.