minimum aberration designs with two level and four level factors summary for optimal assignment of factors to columns of a 4 2 n design we consider extensions of the minimum aberration criterion for the 2 level designs minimum aberration 412n and 422n designs in 16 and 32 runs are obtained and tabulated some results on the wordlength patterns and design classification are obtained

regular fractional factorial designs with minimum aberration and maximum estimation capacity using the approach of finite projective geometry we make a systematic study of estimation capacity a criterion of model robustness under the absence of interactions involving three or more factors some general results providing designs with maximum estimation capacity are obtained in particular for two level factorials it is seen that constructing a design with maximum estimation capacity calls for choosing points from a finite projective geometry such that the number of lines is maximized and the distribution of these lines among the chosen points is as uniform as possible we also explore the connection with minimum aberration designs under which the sizes of the alias sets of two factor interactions which are not aliased with main effects are the most uniform possible

spatial search by quantum walk is optimal for almost all graphs the problem of finding a marked node in a graph can be solved by the spatial search algorithm based on continuous time quantum walks ctqw however this algorithm is known to run in optimal time only for a handful of graphs in this work we prove that for erdos renyi random graphs i e graphs of n vertices where each edge exists with probability p search by ctqw is almost surely optimal as long as p log 3 2 n n consequently we show that quantum spatial search is in fact optimal for almost all graphs meaning that the fraction of graphs of n vertices for which this optimality holds tends to one in the asymptotic limit we obtain this result by proving that search is optimal on graphs where the ratio between the second largest and the largest eigenvalue is bounded by a constant smaller than 1 finally we show that we can extend our results on search to establish high fidelity quantum communication between two arbitrary nodes of a random network of interacting qubits namely to perform quantum state transfer as well as entanglement generation our work shows that quantum information tasks typically designed for structured systems retain performance in very disordered structures

connectivity is a poor indicator of fast quantum search a randomly walking quantum particle evolving by schrodinger s equation searches on d dimensional cubic lattices in o n time when d 5 and with progressively slower runtime as d decreases this suggests that graph connectivity including vertex edge algebraic and normalized algebraic connectivities is an indicator of fast quantum search a belief supported by fast quantum search on complete graphs strongly regular graphs and hypercubes all of which are highly connected in this letter we show this intuition to be false by giving two examples of graphs for which the opposite holds true one with low connectivity but fast search and one with high connectivity but slow search the second example is a novel two stage quantum walk algorithm in which the walking rate must be adjusted to yield high search probability

optimal design with many blocking factors designs for sets of experimental units with many blocking factors are studied it is shown that if the set of blocking factors satisfies a certain simple condition then the information matrix for the design has a simple form in consequence a design is optimal if it is optimal with respect to one particular blocking factor and regular with respect to all the rest in a sense which is made precise in the paper this encompasses several previous results for optimal designs with more than one blocking factor and applications to many other situations are given

some projection properties of orthogonal arrays the definition of an orthogonal array imposes an important geometric property the projection of an oa 2 t 2 k t a 2 t run orthogonal array with k two level factors and strength t onto any t factors consists of copies of the complete 2 t factorial in this article projections of an oa n 2 k t onto t 1 and t 2 factors are considered the projection onto any t 1 factors must be one of three types one or more copies of the complete 2 t 1 factorial one or more copies of a half replicate of 2 t 1 or a combination of both it is also shown that for k t 2 only when n is a multiple of 2 t 1 can the projection onto some t 1 factors be copies of a half replicate of 2 t 1 therefore if n is not a multiple of 2 t 1 then the projection of an oa n 2 k t with k t 2 onto any t 1 factors must contain at least one complete 2 t 1 factorial some properties of projections onto t 2 factors are established and are applied to show that if n is not a multiple of 8 then for any oa n 2 k 2 with k 4 the projection onto any four factors has the property that all the main effects and two factor interactions of these four factors are estimable when the higher order interactions are negligible

selecting 2m p designs using a minimum aberration criterion when some two factor interactions are important we consider the problem of selecting appropriate 2m p designs when some two factor interactions are important current methods in the literature select designs that permit estimation of the postulated model consisting of the main effects and important two factor interactions under the assumption that all of the other effects are negligible when the effects not in the postulated model are not negligible they will bias the estimates of the effects in the model to minimize the contamination of these nonnegligible effects on the model we propose and study a minimum aberration criterion we then discuss the application of this new aberration criterion to compromise plans finally we examine how to search for the best designs according to the criterion and present some results for designs of 16 and 32 runs

strong orthogonal arrays and associated latin hypercubes for computer experiments this paper introduces constructs and studies a new class of arrays called strong orthogonal arrays as suitable designs for computer experiments a strong orthogonal array of strength t enjoys better space filling properties than a comparable orthogonal array in all dimensions lower than t while retaining the space filling properties of the latter in t dimensions latin hypercubes based on strong orthogonal arrays of strength t are more space filling than comparable orthogonal array based latin hypercubes in all g dimensions for any 2 g t 1 copyright 2013 oxford university press

indicator function and its application in two level factorial designs a two level factorial design can be uniquely represented by a polynomial indicator function therefore properties of factorial designs can be studied through their indicator functions this paper shows that the indicator function is an effective tool in studying two level factorial designs the indicator function is used to generalize the aberration criterion of a regular two level fractional factorial design to all two level factorial designs an important identity of generalized aberration is proved the connection between a uniformity measure and aberration is also extended to all two level factorial designs

blocked nonregular two level factorial designs this article discusses the optimal blocking criteria for nonregular two level designs we extend the optimal blocking criteria of cheng and wu to nonregular designs by adapting the g and g2 minimum aberration criteria discussed by tang and deng to define word length pattern for nonregular designs we extend the notion of word to nonregular designs through a polynomial representation of factorial designs we define treatment resolution and block resolution for evaluating the degrees of aliasing and confounding we propose four new criteria which we use to search for optimal blocking schemes of 12 run 16 run and 20 run two level orthogonal arrays

geometric isomorphism and minimum aberration for factorial designs with quantitative factors factorial designs have broad applications in agricultural engineering and scientific studies in constructing and studying properties of factorial designs traditional design theory treats all factors as nominal however this is not appropriate for experiments that involve quantitative factors for designs with quantitative factors level permutation of one or more factors in a design matrix could result in different geometric structures and thus different design properties in this paper indicator functions are introduced to represent factorial designs a polynomial form of indicator functions is used to characterize the geometric structure of those designs geometric isomorphism is defined for classifying designs with quantitative factors based on indicator functions a new aberration criteria is proposed and some minimum aberration designs are presented

some hidden projection properties of orthogonal arrays with strength three summary it is shown that a two level orthogonal array with strength three whose run size is not a multiple of 16 has the hidden projection property that in its projection on to any five factors all the main effects and two factor interactions are estimable if the higher order interactions are negligible the result applies to the fold over of any strength two orthogonal array whose run size is not a multiple of 8 this extends diamond s 1995 result on the fold over of the 12 run plackett burman design some connections with search designs are also presented

spectral similarity for barabasi albert and chung lu models abstract in the paper we have analyzed spectral similarity between barabasi albert and chung lu models we have shown the similarity of spectral distribution for sufficiently large barabasi albert parameter value contrary extreme eigenvalues and principal eigenvector are not similar for those model we provide applications of obtained results related to the spectral graph theory and efficiency of quantum spatial search

effects of independently altering body weight and mass on the energetic cost of a human running model the mechanisms underlying the metabolic cost of running and legged locomotion in general remain to be well understood prior experimental studies show that the metabolic cost of human running correlates well with the vertical force generated to support body weight the mechanical work done and changes in the effective leg stiffness further previous work shows that the metabolic cost of running decreases with decreasing body weight increases with increasing body weight and mass and does not significantly change with changing body mass alone in the present study we seek to uncover the basic mechanism underlying this existing experimental data we find that an actuated spring mass mechanism representing the effective mechanics of human running provides a mechanistic explanation for the previously reported changes in the metabolic cost of human running if the dimensionless relative leg stiffness effective stiffness normalized by body weight and leg length is regulated to be constant the model presented in this paper provides a mechanical explanation for the changes in metabolic cost due to changing body weight and mass which have been previously measured experimentally and highlights the importance of active leg stiffness regulation during human running

space filling fractional factorial designs fractional factorial designs are widely used in various scientific investigations and industrial applications level permutation of factors could alter their geometrical structures and statistical properties this article studies space filling properties of fractional factorial designs under two commonly used space filling measures discrepancy and maximin distance when all possible level permutations are considered the average discrepancy is expressed as a linear combination of generalized word length pattern for fractional factorial designs with any number of levels and any discrepancy defined by a reproducing kernel generalized minimum aberration designs are shown to have good space filling properties on average in terms of both discrepancy and distance several novel relationships between distance distribution and generalized word length pattern are derived it is also shown that level permutations can improve space filling properties for many existing saturated designs a two step construction pro

theory of j characteristics for fractional factorial designs and projection justification of minimum g2 aberration deng tang 1999 introduced the generalised resolution and minimum g aberration criteria for assessing nonregular fractional factorials in tang deng 1999 a relaxed variant of minimum g aberration called minimum g 2 aberration is proposed and studied these criteria are defined using a set of j values called j characteristics in this paper we show that a factorial design is uniquely determined by its j characteristics just as a regular factorial design is uniquely determined by its defining relation the theorem is given through an explicit formula that relates the set of design points to that of j characteristics through this formula projection justification of minimum g 2 aberration is established

nearly orthogonal arrays with mixed levels and small runs in running a factorial experiment it may be desirable to use an orthogonal array with different mixed numbers of factor levels because of the orthogonality requirement such arrays may have a large run size by slightly sacrificing the orthogonality requirement we can obtain nearly orthogonal arrays with economic run size some general methods for constructing such arrays are given for 12 18 20 and 24 runs many orthogonal arrays and nearly orthogonal arrays with mixed levels are constructed and tabulated effects of near orthogonality on estimation efficiency and analysis are studied

comment on a model for the calculation of mechanical power during distance running abstract previous studies have found widely varying estimates of mechanical power for constant velocity running and corresponding widely varying efficiency ratios these results are interpreted as pointing to a need for further research before confidence in measured mechanical power could be established however newtonian mechanics applied to external power together with data on running athletes show that almost as wide a variation is within normal limits for athletes concern should therefore rather be focussed on the measurement of metabolic power and in the procedure of efficiency calculation by division

small composite designs small second order composite designs were suggested by hartley 1959 westlake 1965 provided even smaller designs for k 5 7 and 9 factors for which intricate construction methods were needed here simple designs formed using plackett and burman 1946 designs are offered for k 5 7 and 9 designs with one run fewer than westlake s for k 5 and 7 and three fewer for k 9 are feasible by deleting repeat points that occur in some of the designs

k circulant supersaturated designs a class of supersaturated designs called k circulant designs is explored these designs are constructed from cyclic generators by cycling k elements at a time the class of designs includes many es2 optimal designs some of which are already known and some of which are more efficient than known designs for model estimation under factor sparsity generators for the most efficient designs are listed and projection properties of some of the designs are explored we also illustrate that some k circulant supersaturated designs can be augmented with interaction columns to produce efficient designs for a larger number of factors or for estimating interactions

two level orthogonal screening designs with 24 28 32 and 36 runs abstractthe potential of two level orthogonal designs to fit models with main effects and two factor interaction effects is commonly assessed through the correlation between contrast vectors involving these effects we study the complete catalog of nonisomorphic orthogonal two level 24 run designs involving 3 23 factors and we identify the best few designs in terms of these correlations by modifying an existing enumeration algorithm we identify the best few 28 run designs involving 3 14 factors and the best few 36 run designs in 3 18 factors as well based on a complete catalog of 7570 designs with 28 runs and 27 factors we also seek good 28 run designs with more than 14 factors finally starting from a unique 31 factor design in 32 runs that minimizes the maximum correlation among the contrast vectors for main effects and two factor interactions we obtain 32 run designs that have low values for this correlation to demonstrate the added value of our work we provide a detailed comparison of our desi

general weighted optimality of designed experiments the standard approach to finding optimal experimental designs employs conventional measures of design efficacy such as the a e and d criterion that assume equal interest in all estimable functions of model parameters this paper develops a general theory for weighted optimality allowing precise design selection according to expressed relative interest in different functions in the estimation space the approach employs a very general class of matrix specified weighting schemes that produce easily interpretable weighted optimality criteria in particular for any set of estimable functions and any selected corresponding weights analogs of standard optimality criteria are found that guide design selection according to the weighted variances of estimators of those particular functions the results are applied to solve the a optimal design problem for baseline factorial effects in unblocked experiments

on the existence of saturated and nearly saturated asymmetrical orthogonal arrays we develop a combinatorial condition necessary for the existence of a saturated asymmetrical orthogonal array of strength 2 this condition limits the choice of integral solutions to the system of equations in the bose bush approach and can thus strengthen considerably the bose bush approach as applied to a symmetrical part of such an array as a consequence several nonexistence results follow for saturated and nearly saturated orthogonal arrays of strength 2 one of these leads to a partial settlement of an issue left open in a paper by wu zhang and wang nonexistence of a class of saturated asymmetrical orthogonal arrays of strength 4 is briefly discussed

projections of definitive screening designs by dropping columns selection and evaluation abstract definitive screening designs permit the study of many quantitative factors in a few runs more than twice the number of factors in practical applications researchers often require a desig

2n l designs with weak minimum aberration since not all 2 n l fractional factorial designs with maximum resolution are equally good fries and hunter introduced the minimum aberration criterion for selecting good 2 n l fractional factorial designs with the same resolution we modify the concept of minimum aberration and define weak minimum aberration and show the usefulness of this new design concept using some techniques from finite geometry we construct 2 n l fractional factorial designs of resolution iii with weak minimum aberration further several families of 2 n l fractional factorial designs of resolution iii and iv with minimum aberration are obtained

uniform designs limit aliasing summary when fitting a linear regression model to data aliasing can adversely affect the estimates of the model coefficients and the decision of whether or not a term is significant optimal experimental designs give efficient estimators assuming that the true form of the model is known while robust experimental designs guard against inaccurate estimates caused by model misspecification although it is rare for a single design to be both maximally efficient and robust it is shown here that uniform designs limit the effects of aliasing to yield reasonable efficiency and robustness together aberration and resolution measure how well fractional factorial designs guard against the effects of aliasing here it is shown that the definitions of aberration and resolution may be generalised to other types of design using the discrepancy

design selection and classification for hadamard matrices using generalized minimum aberration criteria deng and tang 1999 and tang and deng 1999 proposed and justified two criteria of generalized minimum aberration for general two level fractional factorial designs the criteria are defined using a set of values called j characteristics in this article we examine the practical use of the criteria in design selection specifically we consider the problem of classifying and ranking designs that are based on hadamard matrices a theoretical result on j characteristics is developed to facilitate the computation the issue of design selection is further studied by linking generalized aberration with the criteria of efficiency and estimation capacity our studies reveal that generalized aberration performs quite well under these familiar criteria

an approach to the construction of asymmetrical orthogonal arrays abstract use of asymmetrical orthogonal arrays oa s for planning industrial experiments with mixed levels has become increasingly popular in addition to applications to experimental investigations in many disciplines they can also be used in the balanced repeated replications method of inference from general stratified survey samples in this article we propose a general approach to the construction of asymmetrical oa s with economic run size and flexibility in the choice of factor levels as applications of this approach we construct several general classes of arrays that include numerous existing classes of arrays as special cases a large number of known asymmetrical oa s can be reproduced by our approach in a unified and simple manner many new arrays are obtained a list of asymmetrical oa s with moderate run size is given our approach consists of three steps we first construct an oa as the generalized kronecker sum of an oa and some difference matrices we then add to the constructed oa co

optimal projective three level designs for factor screening and interaction detection orthogonal arrays oas are widely used in industrial experiments for factor screening suppose that only a few of the factors in the experiments turn out to be important an oa can be used not only for screening factors but also for detecting interactions among a subset of active factors in this article a set of optimality criteria is proposed to assess the performance of designs for factor screening projection and interaction detection and a three step approach is proposed to search for optimal designs combinatorial and algorithmic construction methods are proposed for generating new designs permutations of levels are used for improving the eligibility and estimation efficiency of the projected designs the techniques are then applied to search for best three level designs with 18 and 27 runs many new efficient and practically useful nonregular designs are found and their properties are discussed

mechanical power in running a comparison of different approaches the purposes of the present study were 1 to compare four different methods of calculating mechanical power in running on the basis of comparable data over a wide range of running velocity 2 to examine the linearity of the relation between mechanical power as calculated with the four methods and running velocity eight runners participated in the investigation height 1 82 0 03 m body mass 81 05 4 69 kg a kistler force platform registered all components of the ground reaction force 1000 hz during one foot ground contact which was additionally video taped using two high speed video cameras running at 120 hz four different methods were used to calculate mechanical power two methods determined the mechanical power due to the work done on the athletes center of mass and two were calculated from the motion of the athletes segments the four different methods provided different relations between mechanical power and running velocity the calculations on the basis of kinematic data cannot be recommended to determine efficiency of movement the methods based on ground reaction force measurements revealed significant linear relations r 0 90 r2 0 84 between running velocity and mechanical power

uniform design theory and application a uniform design ud seeks design points that are uniformly scattered on the domain it has been popular since 1980 a survey of ud is given in the first portion the fundamental idea and construction method are presented and discussed and examples are given for illustration it is shown that ud s have many desirable properties for a wide variety of applications furthermore we use the global optimization algorithm threshold accepting to generate ud s with low discrepancy the relationship between uniformity and orthogonality is investigated it turns out that most ud s obtained here are indeed orthogonal

a bayesian approach to the design and analysis of fractionated experiments specifying a prior distribution for the large number of parameters in the statistical model is a critical step in a bayesian approach to the design and analysis of experiments this article shows that the prior distribution can be induced from a functional prior on the underlying transfer function the functional prior requires specification of only a few hyperparameters and thus can be easily implemented in practice the usefulness of the approach is demonstrated through the analysis of some experiments the article also proposes a new class of design criteria and establishes their connections with the minimum aberration criterion

permuting regular fractional factorial designs for screening quantitative factors fractional factorial designs are widely used in screening experiments they are often chosen by the minimum aberration criterion which regards factor levels as symbols for designs with quantitative factors however permuting the levels for one or more factors could alter their geometrical structures and statistical properties we provide a justification of the minimum aberration criterion for quantitative factors and study level permutations for regular fractional factorial designs in order to improve their efficiency for screening quantitative factors we show how regular designs can be linearly permuted to reduce contamination of nonnegligible interactions on the estimation of linear effects without increasing the run size we further show that such linear permutations are unique under the minimum aberration criterion and the best level permutations can be determined without an exhaustive search we establish additional theoretical results for three level designs and obtain the best level permutations for regular designs with 27 and 81 runs we illustrate the practical benefits of level permutation with an antiviral drug combination experiment

designs of variable resolution prior information or background knowledge may suggest that interactions arise only within certain factors when such knowledge is available we propose using a new class of designs designs of variable resolution several constructions are presented statistical justifications for using such designs from minimum g 2 aberration and design efficiency perspectives are provided copyright 2012 oxford university press

majorization framework for balanced lattice designs this paper aims to generalize and unify classical criteria for comparisons of balanced lattice designs including fractional factorial designs supersaturated designs and uniform designs we present a general majorization framework for assessing designs which includes a stringent criterion of majorization via pairwise coincidences and flexible surrogates via convex functions classical orthogonality aberration and uniformity criteria are unified by choosing combinatorial and exponential kernels a construction method is also sketched out

nonregular designs with desirable projection properties in many industrial applications the experimenter is interested in estimating some of the main effects and two factor interactions in this article we rank two level orthogonal designs based on the number of estimable models containing a subset of main effects and their associated two factor interactions by ranking designs in this way the experimenter can directly assess the usefulness of the experimental plan for the purpose in mind we apply the new ranking criterion to the class of all non isomorphic two level orthogonal designs with 16 and 20 runs and introduce a computationally efficient algorithm based on two theoretical results which will aid in finding designs with larger run sizes catalogs of useful designs with 16 20 24 and 28 runs are presented

minimum aberration and model robustness for two level fractional factorial designs summary the performance of minimum aberration two level fractional factorial designs is studied under two criteria of model robustness simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived it is also shown that a minimum aberration design of resolution ilil or higher maximizes the number of two factor interactions which are not aliases of main effects and subject to that condition minimizes the sum of squares of the sizes of alias sets of two factor interactions this roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform it follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here examples are given to show that minimum aberration designs are indeed highly efficient

generalized minimum aberration for asymmetrical fractional factorial designs by studying treatment contrasts and anova models we propose a generalized minimum aberration criterion for comparing asymmetrical fractional factorial designs the criterion is independent of the choice of treatment contrasts and thus model free it works for symmetrical and asymmetrical designs regular and nonregular designs in particular it reduces to the minimum aberration criterion for regular designs and the minimum g 2 aberration criterion for two level nonregular designs in addition by exploring the connection between factorial design theory and coding theory we develop a complementary design theory for general symmetrical designs which covers many existing results as special cases

moment aberration projection for nonregular fractional factorial designs nonregular fractional factorial designs such as plackett burman designs are widely used in industrial experiments for run size economy and flexibility a novel criterion called moment aberration projection is proposed to rank and classify nonregular designs it measures the goodness of a design through moments of the number of coincidences between the rows of its projection designs the new criterion is used to rank and classify designs of 16 20 and 27 runs examples are given to illustrate that the ranking of designs is supported by other design criteria

a general criterion for factorial designs under model uncertainty motivated by two industrial experiments in which rather extreme prior knowledge was used to choose the design we show that the qb criterion which aims to improve the estimation in as many models as possible by incorporating experimenters prior knowledge along with an approximation to the as criterion is more general and has a better statistical interpretation than many standard criteria the generalization and application of the criterion to different types of designs are presented the relationships between qb and other criteria for different situations are explored it is shown that the e s2 criterion is a special case of qb and several aberration type criteria are limiting cases of our criterion so that qb provides a bridge between alphabetic optimality and aberration the two case studies illustrate the potential benefits of the qb criterion r programs for calculating qb are available online as supplemental materials

incorporating prior information in optimal design for model selection an important use of experimental designs is in screening in which experimenters seek to identify significant effects both main effects and potentially interactions from a large set of candidate effects this article goes further than identification of effects introducing a design criterion that seeks to maximize the ability to discriminate between models motivated by the work of meyer steinberg and box the bayesian criterion is based on the hellinger distance between predictive distributions under competing models a bound for the criterion is obtained greatly improving interpretability the set of all possible models to compare is huge and not all models are equally plausible this challenge is addressed through prior distributions on the space of models that indicate preference for intuitively appealing models such as those with few effects more low order than high order effects and inheritance structure between active main effects and interactions techniques for evaluating the criterion a

a trigonometric approach to quaternary code designs with application to one eighth and one sixteenth fractions the study of good nonregular fractional factorial designs has received significant attention over the last two decades recent research indicates that designs constructed from quaternary codes qc are very promising in this regard the present paper shows how a trigonometric approach can facilitate a systematic understanding of such qc designs and lead to new theoretical results covering hitherto unexplored situations we focus attention on one eighth and one sixteenth fractions of two level factorials and show that optimal qc designs often have larger generalized resolution and projectivity than comparable regular designs moreover some of these designs are found to have maximum projectivity among all designs

two level designs to estimate all main effects and two factor interactions we study the design of two level experiments with n runs and n factors large enough to estimate the interaction model which contains all the main effects and all the two factor interactions yet an effect hierarchy assumption suggests that main effect estimation should be given more prominence than the estimation of two factor interactions orthogonal arrays oas favor main effect estimation however complete enumeration becomes infeasible for cases relevant for practitioners we develop a partial enumeration procedure for these cases and we establish upper bounds on the d efficiency for the interaction model based on arrays that have not been generated by the partial enumeration we also propose an optimal design procedure that favors main effect estimation designs created with this procedure have smaller d efficiencies for the interaction model than d optimal designs but standard errors for the main effects in this model are improved generated oas for 7 10 factors and 32 72 runs are smaller or have a higher d efficiency than the smallest oas from the literature designs obtained with the new optimal design procedure or strength 3 oas which have main effects that are not correlated with two factor interactions are recommended if main effects unbiased by possible two factor interactions are of primary interest d optimal designs are recommended if interactions are of primary interest supplementary materials for this article are available online 2017 american statistical association and the american society for quality

blocked regular fractional factorial designs with minimum aberration this paper considers the construction of minimum aberration ma blocked factorial designs based on coding theory the concept of minimum moment aberration due to xu 2003 for unblocked designs is extended to blocked designs the coding theory approach studies designs in a row wise fashion and therefore links blocked designs with nonregular and supersaturated designs a lower bound on blocked wordlength pattern is established it is shown that a blocked design has ma if it originates from an unblocked ma design and achieves the lower bound it is also shown that a regular design can be partitioned into maximal blocks if and only if it contains a row without zeros sufficient conditions are given for constructing ma blocked designs from unblocked ma designs the theory is then applied to construct ma blocked designs for all 32 runs 64 runs up to 32 factors and all 81 runs with respect to four combined wordlength patterns

projective properties of certain orthogonal arrays summary a question of importance in factor screening is when a two level orthogonal design for a multifactor experiment can be projected into lower dimension typically p 2 or 3 new results relate to the projectivity p of saturated designs in which n 1 factors are tested in n runs it is shown that a design obtained by doubling an n x n orthogonal array is always of projectivity p 2 a twolevel cyclic design is either a factorial array and hence has p 2 or it has p 3 a two level orthogonal design with 4m runs m odd has p 3 in particular these results allow the designs derived by plackett burman 1946 to be categorised in terms of these projective properties

construction of two level nonregular designs of strength three with large run sizes abstracttwo level orthogonal arrays of strength 3 permit the study of the main effects and the two factor interactions of the experimental factors these arrays are classified into regular and nonr

projection properties of plackett and burman designs the projection properties of the 2 r q p fractional factorials are well known and have been used effectively in a number of published examples of experimental investigations the plackett and burman designs also have interesting projective properties knowledge of which allows the experimenter to follow up an initial plackett and burman design with runs that increase the initial resolution for the factors that appear to matter and thus permit efficient separation of effects of interest projections of designs into 2 5 dimensions are discussed and the 12 run case is given in detail a numerical example illustrates the practical uses of these projections

hidden projection properties of some nonregular fractional factorial designs and their applications in factor screening often only a few factors among a large pool of potential factors are active under such assumption of effect sparsity in choosing a design for factor screening it is important to consider projections of the design onto small subsets of factors cheng showed that as long as the run size of a two level orthogonal array of strength two is not a multiple of 8 its projection onto any four factors allows the estimation of all the main effects and two factor interactions when the higher order interactions are negligible this result applies for example to all plackett burman designs whose run sizes are not multiples of 8 it is shown here that the same hidden projection property also holds for paley designs of sizes greater than 8 even when their run sizes are multiples of 8 a key result is that such designs do not have defining words of length three or four applications of this result to the construction of e s 2 optimal supersaturated designs are also discussed in particular certain designs constructed by using wu s method are shown to be e s 2 optimal the article concludes with some three level designs with good projection properties

projective three level main effects designs robust to model uncertainty this paper is concerned with designing experiments under the assumption that an analysis strategy is used that considers interactions in addition to main effects a criterion which averages an approximation to as efficiency over lower dimensional projections of the design is introduced to compare designs a columnwise design procedure is used to construct three level designs for six factors in 18 runs the projection efficiencies of these designs are explored and compared with designs obtained from the l 18 orthogonal array results are also given for designs with 14 and 17 runs

characterization of minimum aberration 2 sp n k designs in terms of their complementary designs a general result is obtained that relates the word length pattern of a 2 n k design to that of its complementary design by applying this result and using group isomorphism we are able to characterize minimum aberration 2 n k designs in terms of properties of their complementary designs the approach is quite powerful for small values of 2 n k n 1 in particular we obtain minimum aberration 2 n k designs with 2 n k n 1 1 to 11 for any n and k

construction of optimal multi level supersaturated designs author s xu hongquan wu cfj abstract a supersaturated design is a design whose run size is not large enough for estimating all the main effects the goodness of multi level supersaturated designs can be judged by the generalized minimum aberration criterion proposed by xu and wu 2001 optimal supersaturated designs are shown to have a periodic property and general methods for constructing optimal multilevel supersaturated designs are proposed inspired by the addelman kempthorne construction of orthogonal arrays optimal multi level supersaturated designs are given in an explicit form columns are labeled with linear or quadratic polynomials and rows are points over a finite field additive characters are used to study the properties of resulting designs some small optimal supersaturated designs of 3 4 and 5 levels are listed with their properties

some identities on q sp n m designs with application to minimum aberration designs chen and hedayat and tang and wu studied and characterized minimum aberration 2 n m designs in terms of their complemetary designs based on a new and more powerful approach we extend the study to identify minimum aberration q n m designs through their complementary designs by using macwilliams identities and krawtchouk polynomials in coding theory we obtain some general and explicit relationships between the wordlength pattern of a q n m design and that of its complementary design these identities provide a powerful tool for characterizing minimum aberration q n m designs the case of q 3 is studied in more details

simple method for measuring center of mass work during field running abstract the purpose of this study was to propose and validate a new simple method for calculation of center of mass work during field running in order to avoid the use of costly and inconvenient measurement devices this method relies on spring mass model and measurements of average horizontal velocity and contact and flight times during running ten male recreational subjects ran on a dynamometer treadmill at different velocities ranging from 2 22 to 4 44 m s 1 during 4 min 30 s for each velocity twenty consecutive steps were analyzed after 3 min 30 s the potential w pot forward kinetic w kinf and the total center of mass w ext work data obtained with this new method were compared with the reference data calculated from ground reaction force measurements w ext w pot and w kinf values calculated with the proposed method were respectively 3 39 0 77 higher 4 14 0 72 lower and 7 34 1 08 higher than values obtained by the reference method furthermore significant linear regressions close to the identity line were obtained between the reference and the proposed method values of works r 0 99 p w ext r 0 98 p w pot r 0 98 p w kinf it was concluded that this new method could provide a good estimate of center of mass work in field running thanks to a few simple mechanical parameters

an explicit expression for the moment in multibody systems the equations of motion are formulated for a set of interacting rigid bodies an explicit formulation could be derived for the total moment of the forces on such a system this is applied to 1 the intersegmental moment in a multisegment rigid body model and 2 the interpretation of the centre of pressure as defined in the studies of human posture

the decomposition of treatment degrees of freedom in quantitative factorial experiments summary in an experiment in which every treatment factor has the same number p of levels where p is prime there is a classical breakdown of the treatment degrees of freedom into components such as ab ab2 ab2 c each of p 1 degrees of freedom the disadvantages of this system are i the lack of canonical single degrees of freedom and ii the fact that the breakdown bears no relation to the contrasts typically of interest if all treatment factors are quantitative in this note we decompose the treatment degrees of freedom into single degrees of freedom and use these to show how appropriate choice of levels of quantitative factors enables quantitative contrasts to be completely unconfounded in classical designs

blocking in compound dispersion experiments in quality design a dispersion experiment attempts to estimate the effects of control factors on the dispersion introduced by error factors that is the effect of product design on the stability of the performance of the product as its operating environment changes in conducting a dispersion experiment it is important to distinguish the sources of dispersion that one wishes to control from extraneous sources of dispersion so blocking is often necessary this is especially true when the quality or utility of a product is appraised subjectively as in conjoint analysis compound designs for dispersion experiments are orthogonal array designs that generalize taguchi s direct product designs compared to direct product designs compound designs are often stronger for a given size or smaller for a given strength this article discusses blocked compound dispersion experiments providing that blocks form an orthogonal array of strength 1 for the error factors dispersion within blocks may be pooled to estima

optimal fractions of two level factorials under a baseline parameterization two level fractional factorial designs are considered under a baseline parameterization the criterion of minimum aberration is formulated in this context and optimal designs under this criterion are investigated the underlying theory and the concept of isomorphism turn out to be significantly different from their counterparts under orthogonal parameterization and this is reflected in the optimal designs obtained copyright 2012 oxford university press

optimal foldover plans for two level nonregular orthogonal designs this article considers optimal foldover plans for nonregular designs by using the indicator function we define words with fractional lengths the extended word length pattern is then used to select among nonregular foldover designs some general properties of foldover designs are obtained using the indicator function we prove that the full foldover plan that reverses the signs of all factors is optimal for all 12 run and 20 run orthogonal designs the optimal foldover plans for all 16 run regular and nonregular orthogonal designs are constructed and tabulated for practical use optimal foldover plans for higher order orthogonal designs can be constructed in a similar manner

minimum aberration construction results for nonregular two level fractional factorial designs summary nonregular two level fractional factorial designs are designs which cannot be specified in terms of a set of defining contrasts the aliasing properties of nonregular designs can be compared by using a generalisation of the minimum aberration criterion called minimum g2 aberration until now the only nontrivial designs that are known to have minimum g2 aberration are designs for n runs and m n 5 factors in this paper a number of construction results are presented which allow minimum g2 aberration designs to be found for many of the cases with n 16 24 32 48 64 and 96 runs and m n 2 2 factors

generalized resolution for orthogonal arrays the generalized word length pattern of an orthogonal array allows a ranking of orthogonal arrays in terms of the generalized minimum aberration criterion xu and wu 2001 we provide a statistical interpretation for the number of shortest words of an orthogonal array in terms of sums of r values based on orthogonal coding or sums of squared canonical correlations based on arbitrary coding directly related to these results we derive two versions of generalized resolution for qualitative factors both of which are generalizations of the generalized resolution by deng and tang 1999 and tang and deng 1999 we provide a sufficient condition for one of these to attain its upper bound and we provide explicit upper bounds for two classes of symmetric designs factor wise generalized resolution values provide useful additional detail