Parallel Flats Research Articles

Fast construction of efficient two-level parallel flats designs

Two-level parallel flats designs (PFDs) are nonregular designs that retain some of the simplicity of regular fractional factorial designs. PFDs have many desirable properties, such as flexible run sizes and block diagonal information matrices. In this article, we develop a fast algorithm for constructing efficient two-level PFDs, optimizing the parallel flats structure via a novel application of coordinate exchange. The resulting designs fill gaps among existing nonregular designs and outperform many comparably-sized designs in terms of G- and G2-aberration. An efficient implementation of the proposed algorithm in Matlab and GNU Octave can be found in the online Supplementary Material. Several new designs with desirable properties are generated and tabulated.

Structure of Nonregular Two-Level Designs

Two-level fractional factorial designs are often used in screening scenarios to identify active factors. This article investigates the block diagonal structure of the information matrix of nonregular two-level designs. This structure is appealing since estimates of parameters belonging to different diagonal submatrices are uncorrelated. As such, the covariance matrix of the least squares estimates is simplified and the number of linear dependencies is reduced. We connect the block diagonal information matrix to the parallel flats design (PFD) literature and gain insights into the structure of what is estimable and/or aliased using the concept of minimal dependent sets. We show how to determine the number of parallel flats for any given design, and how to construct a design with a specified number of parallel flats. The usefulness of our construction method is illustrated by producing designs for estimation of the two-factor interaction model with three or more parallel flats. We also provide a fuller understanding of recently proposed group orthogonal supersaturated designs. Benefits of PFDs for analysis, including bias containment, are also discussed.

Two-level parallel flats designs

Regular 2n−p designs are also known as single flat designs. Parallel flats designs (PFDs) consisting of three parallel flats (3-PFDs) are the most frequently utilized PFDs, due to their simple structure. Generalizing to f-PFD with f>3 is more challenging. This paper aims to study the general theory for the f-PFD for any f≥3. We propose a method for obtaining the confounding frequency vectors for all nonequivalent f-PFDs, and to find the least G-aberration (or highest D-efficiency) f-PFD constructed from any single flat. PFDs are particularly useful for constructing nonregular fraction, split-plot or randomized block designs. We also characterize the quaternary code design series as PFDs. Finally, we show how designs constructed by concatenating regular fractions from different families may also have a parallel flats structure. Examples are given throughout to illustrate the results.

Design of Edible Oil Degradation Tool by Using Electromagnetic Field Absorbtion Principle which was Characterized to Peroxide Number

The identification of changes in oil quality has been conducted by indicating the change of dielectric constant which was showed by sensor voltage. Sensor was formed from two parallel flats that worked by electromagnetic wave propagation principle. By measuring its amplitude of electromagnetic wave attenuation caused by interaction between edible oil samples and the sensor, dielectric constant could be identified and estimated as well as peroxide number. In this case, the parallel flats were connected to an electric oscillator 700 kHz. Furthermore, sensor system could showed measurable voltage differences for each different samples. The testing carried out to five oil samples after undergoing an oxidation treatment at fix temperature of 235oC for 0, 5, 10, 15 and 20 minutes. Iodometry method testing showed peroxide values about 1.99, 9.95, 5.96, 11.86, and 15.92 meq/kg respectively with rising trend. Besides that, the testing result by sensor system showed voltages values 1.139, 1.147, 1.165, 1.173, and 1.176 volts with rising trend, respectively. It means that the higher sensor voltages showed the higher damage rate of edible oil when the change in sensor voltage was caused by the change in oil dielectric constant in which heating process caused damage in edible oil molecules structure. The more damage of oil structure caused the more difficulties of oil molecules to polarize and it is indicated by smaller dielectric constant. Therefore electric current would be smaller when sensor voltage was higher. On the other side, the higher sensor voltage means the smaller dielectric constant and the higher peroxide number.

Interferometric spatial coherence tomography: focusing fringe contrast to planes of interest using a quasi-monochromatic structured light source

Interferograms of plane parallel optical flats are hard to interpret when acquired with coherent illumination because of the complex fringe pattern resulting from the superposition of three main contributions, namely from the reference surface and the front and back sample surfaces. We illuminate the sample by a field of high temporal and specially tailored partial spatial coherence. This limits the fringe contrast to sheets of adjustable position and thickness along the axis of the interferometer. We outline the technique and demonstrate its application together with phase shifting interferometry to extract the topography of front and back surfaces of transparent samples.

Design and Analysis of Two-Level Factorial Experiments With Partial Replication

In a two-level factorial experiment, we consider construction of parallel-flats designs with two identical parallel flats that allow estimation of a set of specified possibly active effects and the pure error variance. A set of sufficient conditions is presented for the designs to be D-optimal for the specified effects, assuming that the other effects are negligible, over the class of competing parallel-flats designs. In addition, an algorithm is developed to generate the D-optimal designs with a choice of flexible degrees of freedom for the pure error variance. Because the proposed partially replicated designs are highly efficient in estimating the possibly active effects and provide a replication-based estimate of the error variance, they provide a practical compromise between the power in identifying truly active effects and the number of runs in experiments. This property is verified through a simulation study.

Three Parallel Flats Designs for Two‐level Factorial Experiments

This paper investigates the properties of the class of three parallel flats designs for two‐level factorial experiments. It shows that the designs constructed from this class of designs can have a very simple correlation structure. The correlation of any pair of best linear unbiased estimators of factorial effects is 0, ⅓ or ¼. Furthermore, the designs obtained also have high D‐efficiency. Finally, a class of designs is generated with run‐size N= 12 to illustrate the use of the theorem.

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.

Orthogonal factorial design of parallel flats type and its application in biomedical research

In this paper, orthogonal designs of parallel flats type are generated for 3 m and mixed 2 n ×3 m factorial experiments, where factor effects include the general mean, main effects, and certain classes of two-factor interactions. The applications of mixed factorial designs in the PCR process of DNA identification and drug interactions in clinical studies are also discussed. This work will be useful for applied statisticians to implement optimal designs in practice.

Optimal 2n factorial designs of parallel flats type

In this paper, the structure of information matrices are studied for 2n experiments of parallel flats type designs. Some parallel flats type designs are proved to be optimal in certain class of designs with respect to D-optimal and E-optimal criteria. Especially if a design is obtained from an orthogonal design by adding (or deleting) a treatment, the design is optimal over all possible designs with respect to type 1 (or type 2) criteria. These results will be very useful in practice to check the optimality when missing data occur in an orthogonal experiment.

Orthogonal designs of parallel flats type

Consider an s n factorial experiment, where s is such that GF( s) exists. Let T i be the set of s n− r treatments (often called ‘assemblies’ or ‘runs’) t satisfying the equations At = c i over GF( s), where A ( r × n) is of rank r, t is ( n × 1), and c i is an ( r × 1) vector for i = 1, …, f. Let T be obtained by taking all the T i together so that T has N = fs n− r treatments. Let C = ( c 1, …, c f ). Thus, T is determined by the pair ( A, C). Assume that F is any vector of distinct sets of factorial effects, each set (except for μ, the general mean) carrying ( s − 1) degres of freedom. Let M be the information matrix for estimating all the parameters in F by using T, assuming that the effects not included in F are negligible. In this paper, we give a simple necessary and sufficient condition on A and C such that M is diagonal. We also give a large set of examples for the case when s = 2, and F consists of the general mean, main effects, and certain classes of two-factor interactions. (Readers who are interested only in the examples may go directly to Section 3.)

The spectrum of the information matrix for sn parallel flats fractions when s is prime

A parallel flats fraction (PFF) for an s n factorial experiment, when s is a prime or prime power, is defined as the set of all solutions over GF( s) to At = c i , i = 1, 2, …, f where A is r × n of rank r. The fraction is completely determined by A and C = [ c 1, c 2, …, c f ]; hence all optimality conditions are also completely determined by A and C. It has been shown before that for s a prime the information matrix for a parallel flats fraction can be characterized in terms of the direct sum of relatively small matrices over the cyclotomic field of order s over the rationals. In this paper for s a prime we obtain the eigenvalues (the spectrum) of the information matrix and the eigenvectors directly in terms of these smaller matrices over the cyclotomic field. The usefulness of these results are in the construction of optimal fractions and the comparison of competing fractions.

Sequential factorial probing designs for identifying and estimating non-negligible factorial effects for the 2 m experiment under the tree structure

Consider a 2 m factorial experiment. Let A (2 m × 1) be the vector of ‘factorial effects’. Let θ be a v × 1 subvector of A . Then θ is said to have a ‘tree structure’ if it is such that if A( i 1,…, i k ) (which, for all k, denotes the k-factor interaction between factors i 1,…, i k , where 1≤ i 1< i 2< i k ≤ m) belongs to θ , then at least one ( k−1)-factor interaction A( j 1,hellip;, j k−1 ) (where the set ( j 1,hellip;, j k−1 ) is a subset of ( i 1,…, i k )), also belongs to θ . Now, suppose that an element of A is non-negligible if and only if it belongs to θ . Suppose further that θ is unknown (except that it is known to have a tree structure, and also that v is known). In this paper, for m≤4, we obtain sequential designs for identifying and estimating θ (denoted later as L ∗). We employ the intersection sieve and other tentative decision rules introduced in Srivastava (1987). Note that the earlier work deals only with single stage search designs, or with optimal balanced designs or designs of the parallel flats types, the latter two being meant for estimating a given subset of A (under the assumption that the elements of A not in the subset are negligible). Thus, the present work is at a level of generality higher than ever attempted before. Also, it will be useful in signal processing.

Optimal fractional factorial designs for estimating interactions of one factor with all others: 3mSeries

In some experimental situations, only one factor is expected to interact with other factors. Designs which permit estimation of all main effects and the interactions of one factor ‘With All Others’, are termed WAO designs. This paper discusses the existence and construction of sm WAO designs. A series of WAO designs are presented for the 3m factorial, for m = 6, 7, ... , 14. The p non-negligible effects are estimable in 9f∗ runs, where f∗ is the smallest integer such that 9f∗ ≥p. These designs are determinant optimal within the class of parallel flats fractions and, except for the case f∗ = 9, are new. They are ideally suited for sequential experiments.

Free-expansion melting of two dimensional colloidal crystals

Abstract Two-dimensional (2D) melting transitions have been observed in two freely expanding colloidal monolayer crystals formed from 1.01 μm polystyrene microspheres in an aqueous suspension between parallel optical flats, and with expansion rates differing by a factor of six. The data consist of digitized images of the 2D microsphere populations within a fixed sample area made from video records of the entire transition from ordered solid to liquid. The correlation functions computed from these data are consistent with a continuous two-step melting transition. However, direct observations of the time development of defect creation and evolution reveal that the systems may melt via a first order process. Although the melting rates differed appreciably, the results of the two melts are essentially the same.

Free-expansion melting of a colloidal monolayer.

A two-dimensional melting transition has been observed in a freely expanding colloidal monolayer lattice formed of 1.01-\ensuremath{\mu}m polystyrene microspheres in an aqueous suspension between two parallel optical flats. The correlation functions computed from digitized images of the particle distribution within a fixed sampling area are consistent with a continuous two-step melting transition. However, direct observations of defect creation and evolution reveal that the system may melt via a first-order process.

Theory of factorial designs of the parallel flats type. I: the coefficient matrix

Let GF( s ) be the finite field with s elements.(Thus, when s =3, the elements of GF( s ) are 0, 1 and 2.)Let A ( r × n ), of rank r , and c i ( i =1,…, f ), ( r ×1), be matrices over GF( s ). (Thus, for n =4, r =2, f =2, we could have A =[ 1110 0121 ], c 1 =[ 1 0 ], c 2 =[ 0 2 ].) Let T i ( i =1,…, f ) be the flat in EG( n , s ) consisting of the set of all the s n − r solutions of the equations At=c i , where t ′=( t 1 ,…, t n ) is a vector of variables.(Thus, EG(4, 3) consists of the 3 4 =81 points of the form ( t 1 , t 2 , t 3 , t 4 ), where t 's take the values 0,1,2 (in GF(3)). The number of solutions of the equations At = c i is s n−r , where r =Rank( A ), and the set of such solutions is said to form an ( n − r )-flat, i.e. a flat of ( n − r ) dimensions. In our example, both T 1 and T 2 are 2-flats consisting of 3 4−2 =9 points each. The flats T 1 , T 2 ,…, T f are said to be parallel since, clearly, no two of them can have a common point. In the example, the points of T 1 are (1000), (0011), (2022), (0102), (2110), (1121), (2201), (1212) and (0220). Also, T 2 consists of (0002), (2010), (1021), (2101), (1112), (0120), (1200), (0211) and (2222).) Let T be the fractional design for a s n symmetric factorial experiment obtained by taking T 1 , T 2 ,…, T f together. (Thus, in the example, 3 4 =81 treatments of the 3 4 factorial experiment correspond one-one with the points of EG(4,3), and T will be the design (i.e. a subset of the 81 treatments) consisting of the 18 points of T 1 and T 2 enumerated above.) In this paper, we lay the foundation of the general theory of such ‘parallel’ types of designs. We define certain functions of A called the alias component matrices, and use these to partition the coefficient matrix X ( n × v ), occuring in the corresponding linear model, into components X . j ( j =0,1,…, g ), such that the information matrix X is the direct sum of the X ′. j X . j . Here, v is the total number of parameters, which consist of (possibly μ), and a (general) set of (geometric) factorial effects (each carrying ( s −1) degrees of freedom as usual). For j ≠0, we show that the spectrum of X ′. j X . j does not change if we change (in a certain important way) the usual definition of the effects. Assuming that such change has been adopted, we consider the partition of the X . j into the X ij ( i =1,…, f ). Furthermore, the X ij are in turn partitioned into smaller matrices (which we shall here call the) X ijh . We show that each X ijh can be factored into a product of 3 matrices J , ζ (not depending on i,j , and h ) and Q(j,h,i) where both the Kronecker and ordinary product are used. We introduce a ring R using the additive groups of the rational field and GF( s ), and show that the Q(j,h,i) belong to a ring isomorphic to R . When s is a prime number, we show that R is the cyclotomic field. Finally, we show that the study of the X . j and X ′. j X . j can be done in a much simpler manner, in terms of certain relatively small sized matrices over R .

Kinematic accuracy of a flat generating system

SUMMARY The Flat Generating System considered has been used on automatic lathes for many years. Its function is to generate parallel flats, squares, hexagons and other regular patterns of flats on turned components as part of the turning operation. Figure 1 illustrates two typical components with hexagons produced by this System. It has been generally accepted that the system does not produce truly flat sides on the form being generated. However there is no apparent evidence than an analysis of the accuracy of the system has been carried out. This may be because under certain established empirical conditions the errors have not been of sufficient magnitude to warrant attention. A preliminary investigation does reveal that varying degrees of error occur with variations in the machining condition and that at a certain stage a change from a concave error to a convex error takes place. On this basis, there would appear to be merit in examining the system with a view to optimizing the machining conditions to a...

Continuously Tunable Resonant Ruby Laser Reflector

The general theoretical requirements for making multisurfaced (resonant) reflectors frequency-tunable are described. Specifically, eflectors that can be used to stabilize and control the lasing spectrum of ruby lasers are considered. Fabry-Perot ring analysis showed that a four-surface reflector (two parallel sapphire flats) with a spacer (stellite) having close to the ideal thermal expansion coefficient, for thermal frequency tunability, could be used to continuously tune a ruby laser near 6944 A more than 0.3 A with a spectral width less than +/-0.01 A.

Ring Heat Source Probe for Rapid Determination of Thermal Conductivity of Rocks

A ring heat source probe has been developed for the rapid determination of thermal conductivity of rocks. In addition to the short period of time required to run the test (2–3 min), sample preparation is minimal, requiring only that parallel flats approximately two cm in diameter be cut or ground on opposite sides of the sample. A calibration chart can be provided with each probe so that thermal conductivity can be obtained from simple experimental data without elaborate calculation.