Alias Structure Research Articles

The Summary of Effect Aliasing Structure for Two-Level Supersaturated and Factorial Designs

The Summary of Effect Aliasing Structure for Two-Level Supersaturated and Factorial Designs

Alias‐informed model selection (AIMS) for 7 and 8 factor no‐confounding 16‐run fractional factorial designs

AbstractNonregular fractional factorial designs are a preferable alternative to regular resolution IV designs because they avoid confounding 2‐factor interactions. As a result, nonregular designs can estimate and identify a few active 2‐factor interactions. However, due to the sometimes complex alias structure of nonregular designs, standard factor screening strategies can fail to identify all active effects. We report on a screening technique that takes advantage of the alias structure of these nonregular designs. This alias‐informed‐model‐selection (AIMS) technique has been used previously for a specific 6‐factor nonregular design. We show how the AIMS technique can be applied to 7‐ and 8‐factor nonregular designs, completing the exposition of this method for all 16‐run 2‐level designs that are viable alternatives to standard Resolution IV fractional factorial designs. We compare AIMS to three other standard analysis methods for nonregular designs, stepwise regression, the lasso, and the Dantzig selector. AIMS consistently outperforms these methods in identifying the set of active factors.

Aliased informed model selection strategies for six‐factor no‐confounding designs

AbstractNonregular designs are a preferable alternative to regular resolution IV designs because they avoid confounding two‐factor interactions. As a result nonregular designs can estimate and identify a few active two‐factor interactions. However, due to the sometimes complex alias structure of nonregular designs, standard screening strategies can fail to identify all active effects. In this paper, we explore a specific no‐confounding six‐factor 16‐run nonregular design with orthogonal main effects. By utilizing our knowledge of the alias structure, we can inform the model selection process. Our aliased informed model selection (AIMS) strategy is a design‐specific approach that we compare to three generic model selection methods; stepwise regression, Lasso, and the Dantzig selector. The AIMS approach substantially increases the power to detect active main effects and two‐factor interactions versus the aforementioned generic methodologies.

Economic Trend Resistant2n-(n-k) Designs of Resolutions III and IV Based on Hadamard Matrices

This article utilizes the Normalized Sylvester-Hadamard Matrices of size 2 k x2 k and their associated saturated orthogonal arrays OA(2 k , 2 k - 1, 2, 2) topropose analgorithmbased on factor projection (Backward/Forward) for the construction of three systematic run-after-run2 n-(n-k) fractional factorial designs: (i) minimum cost trend free 2 n-(n-k) designsof resolution III (2 k-1 ≤n≤2 k – 1 – k)by backward factor deletion (ii) minimum cost trend free 2 n-(n-k) designsof resolution III (k+1≤n≤ 2 k-1 – 2+k ) by forward factor addition (iii) minimum costtrend free 2 n-(n-k) designsof resolution IV (2 k-2 ≤n≤2 k-1 -2) ,where each 2 n-(n-k) design is economic minimizing the number of factor level changes between the 2 k successive runs and allows for the estimation of all factor main effects unbiased by the linear time trend,which might be present in the 2 k sequentially generated responses. The article gives for each 2 n-(n-k) design: (i) the defining contrast displaying the design’s alias structure(ii) the k independent generators for sequencingthe design’s 2 n-(n-k) runs by the Generalized Fold over Scheme and (ii) the minimum total cost of factor level changes between the 2 n-(n-k) runs of the design. Proposed designs compete well with existing systematic2 n-(n-k) designs (of either resolution) in minimizing the experimental costandin securing factors’ resistance to the non-negligible time trend. Keywords : Sequential fractional factorial experimentation; Time trend free systematic run orders; Generalizedfoldover scheme for sequencing experimentalruns; The total cost of factor level changes between successive runs; The Normalized Sylvester –Hadamard Matrices; Orthogonal Arrays and factor projection; Design resolution and the alias structure. DOI: 10.7176/JEP/11-25-05 Publication date: September 30 th 2020

Improving Innovation from Science Using Kernel Tree Methods as a Precursor to Designed Experimentation

A key challenge in applied science when planning a designed experiment is to determine the aliasing structure of the interaction effects and selecting the appropriate levels for the factors. In this study, kernel tree methods are used as precursors to identify significant interactions and levels of the factors useful for developing a designed experiment. This approach is aligned with integrating data science with the applied sciences to reduce the time from innovation in research and development to the advancement of new products, a very important consideration in today’s world of rapid advancements in industries such as pharmaceutical, medicine, aerospace, etc. Significant interaction effects for six common independent variables using boosted trees and random forests of k = 1000 and k = 10,000 bootstraps were identified from industrial databases. The four common variables were related to speed, pressing time, pressing temperature, and fiber refining. These common variables maximized tensile strength of medium density fiberboard (MDF) and the ultimate static load of oriented strand board (OSB), both widely-used industrial products. Given the results of the kernel tree methods, four possible designs with interaction effects were developed: full factorial, fractional factorial Resolution IV, Box–Behnken, and Central Composite Designs (CCD).

SANTIS: Sampling-Augmented Neural neTwork with Incoherent Structure for MR image reconstruction.

To develop and evaluate a novel deep learning-based reconstruction framework called SANTIS (Sampling-Augmented Neural neTwork with Incoherent Structure) for efficient MR image reconstruction with improved robustness against sampling pattern discrepancy. With a combination of data cycle-consistent adversarial network, end-to-end convolutional neural network mapping, and data fidelity enforcement for reconstructing undersampled MR data, SANTIS additionally utilizes a sampling-augmented training strategy by extensively varying undersampling patterns during training, so that the network is capable of learning various aliasing structures and thereby removing undersampling artifacts more effectively and robustly. The performance of SANTIS was demonstrated for accelerated knee imaging and liver imaging using a Cartesian trajectory and a golden-angle radial trajectory, respectively. Quantitative metrics were used to assess its performance against different references. The feasibility of SANTIS in reconstructing dynamic contrast-enhanced images was also demonstrated using transfer learning. Compared to conventional reconstruction that exploits image sparsity, SANTIS achieved consistently improved reconstruction performance (lower errors and greater image sharpness). Compared to standard learning-based methods without sampling augmentation (e.g., training with a fixed undersampling pattern), SANTIS provides comparable reconstruction performance, but significantly improved robustness, against sampling pattern discrepancy. SANTIS also achieved encouraging results for reconstructing liver images acquired at different contrast phases. By extensively varying undersampling patterns, the sampling-augmented training strategy in SANTIS can remove undersampling artifacts more robustly. The novel concept behind SANTIS can particularly be useful for improving the robustness of deep learning-based image reconstruction against discrepancy between training and inference, an important, but currently less explored, topic.

Matrix Image Method for Ranking Nonregular Fractional Factorial Designs

Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. In this paper, we study matrix image theory and present a new method for distinguishing and assessing nonregular designs with complex alias structure, which works for all symmetrical and asymmetrical, regular and nonregular orthogonal arrays. Based on the matrix image theory, our proposed method captures orthogonality and projection properties. Empirical studies show that the proposed method has a more precise differentiation capacity when comparing with some other criteria.

Fourth order Galilean invariance for the lattice Boltzmann method

Using the structure of a recursive asymptotic analysis we derive conditions on cumulants that guarantee a prescribed order of Galilean invariance for lattice Boltzmann models. We then apply these conditions to three different lattice Boltzmann models and obtain three models with fourth order accurate advection. One of the models uses 27 speeds on a body centered cubic lattice, one uses 33 speeds on an extended Cartesian lattice and one uses 27 speeds plus three finite differences on a Cartesian lattice. All models offer too few degrees of freedom to impose the conditions on the cumulants directly. However, the specific aliasing structure of these lattices permit fourth order accuracy for a model specific optimal reference temperature. Our theoretical derivations are confirmed by measuring the phase lag of traveling vortexes and shear waves.

A blue noise pattern sampling method based on cloud computing to prevent aliasing

The high frequency of the image through pre-filtering and sampling cannot be eliminated, whereby the power spectrum of the oscillation may appear the aliasing phenomenon, the sampling scheme based on cloud computing proposed two standard blue noise patterns: step blue noise and unimodal blue noise. However, a large number of sampling points usually results in large processing requirements. In this paper we propose an object-order algorithm by using an octree and n-bit quantised gray, MIP average complexity can be reduced to O (nˆ2). This improvement makes the interactive visualisation and the data storage security of MIP greatly improved in large capacity data application. Experimental results show that the low sampling rate model based on cloud computing can effectively prevent aliasing structure, in a high sampling rate model based on cloud computing also perform equally well. Simulation results employing H.264's redundant slice mechanism show significant performance gains over conventional error-resilient encoding methods and native redundant encoding methods.

A blue noise pattern sampling method based on cloud computing to prevent aliasing

The high frequency of the image through pre-filtering and sampling cannot be eliminated, whereby the power spectrum of the oscillation may appear the aliasing phenomenon, the sampling scheme based on cloud computing proposed two standard blue noise patterns: step blue noise and unimodal blue noise. However, a large number of sampling points usually results in large processing requirements. In this paper we propose an object-order algorithm by using an octree and n-bit quantised gray, MIP average complexity can be reduced to O (nˆ2). This improvement makes the interactive visualisation and the data storage security of MIP greatly improved in large capacity data application. Experimental results show that the low sampling rate model based on cloud computing can effectively prevent aliasing structure, in a high sampling rate model based on cloud computing also perform equally well. Simulation results employing H.264's redundant slice mechanism show significant performance gains over conventional error-resilient encoding methods and native redundant encoding methods.

Destroying Aliases from the Ground and Space: Super-Nyquist ZZ Cetis in K2 Long Cadence Data

Abstract With typical periods of the order of 10 minutes, the pulsation signatures of ZZ Ceti variables (pulsating hydrogen-atmosphere white dwarf stars) are severely undersampled by long-cadence (29.42 minutes per exposure) K2 observations. Nyquist aliasing renders the intrinsic frequencies ambiguous, stifling precision asteroseismology. We report the discovery of two new ZZ Cetis in long-cadence K2 data: EPIC 210377280 and EPIC 220274129. Guided by three to four nights of follow-up, high-speed ( s) photometry from the McDonald Observatory, we recover accurate pulsation frequencies for K2 signals that reflected four to five times off the Nyquist with the full precision of over 70 days of monitoring (∼0.01 μHz). In turn, the K2 observations enable us to select the correct peaks from the alias structure of the ground-based signals caused by gaps in the observations. We identify at least seven independent pulsation modes in the light curves of each of these stars. For EPIC 220274129, we detect three complete sets of rotationally split (dipole mode) triplets, which we use to asteroseismically infer the stellar rotation period of 12.7 ± 1.3 hr. We also detect two sub-Nyquist K2 signals that are likely combination (difference) frequencies. We attribute our inability to match some of the K2 signals to the ground-based data to changes in pulsation amplitudes between epochs of observation. Model fits to SOAR spectroscopy place both EPIC 210377280 and EPIC 220274129 near the middle of the ZZ Ceti instability strip, with K and 11810 ± 210 K, and masses 0.57 ± 0.03 M ⊙ and 0.62 ± 0.03 M ⊙, respectively.

Using Definitive Screening Designs to Identify Active First- and Second-Order Factor Effects

Definitive screening designs (DSDs) were recently introduced by Jones and Nachtsheim (2011b). The use of three-level factors and the desirable aliasing structure of the DSDs make them potentially suitable for identifying main effects and second-order terms in one stage of experimentation. However, as the number of active effects approaches the number of runs, the performance of standard model-selection routines will inevitably degrade. In this paper, we characterize the ability of DSDs to correctly identify first- and second-order model terms as a function of the level of sparsity, the number of factors in the design, the signal-to-noise ratio, the model type (unrestricted or following strong heredity), the model-selection technique, and the number of augmented runs. We find that minimum-run-size DSDs can be used to identify active terms with high probability as long as the number of effects is less than or equal to about half the number of runs and the signal-to-noise ratios for the active effects are above about 2.0. We also find that if minimum-run-size designs are augmented with four or more runs, the number of model terms that can be identified with high probability increases substantially. Among the model-selection methods investigated, we found that both Lasso and the Gauss–Dantzig selector (both based on AICc) can be used to effectively identify active model terms in the presence of unrestricted models. For models following strong heredity, the SHIM method developed by Choi et al. (2010) was the best among methods tested that were designed for the strong-heredity case.

A Bayesian approach to the identification of active location and dispersion factors

ABSTRACTIn this article, we extend the modified Box–Meyer method and propose an approach to identify both active location and dispersion factors in a screening experiment. Since several candidate models can be simultaneously considered under the framework of Bayesian model averaging, the proposed method can overcome the problem of missing the identification of some active factors caused by either the alias structure or misspecification of the location model. For illustration, three practical experiments and one synthetic data set are analyzed.

Minimum Cost Linear Trend Free 2n-(n-k) Designs of Resolution IV

This article extends the resolution of time trend free designs for sequential 2n-p experiments from III into IV and minimizes the number of factor level changes between runs (i.e., cost) by constructing a catalog of (2k−2 −1) minimum cost linear trend free resolution IV 2n−(n−k) designs (2k−2 ≤ n ≤ 2k−1−2) from the full 2k factorial experiment using the interactions-main effects assignment technique. Each systematic 2n−(n−k) design in the catalog is economic in minimum number of factor level changes and allows for the estimation of all n main effects unbiased by either the linear time trend (which may be present in the 2n−(n−k) sequentially generated responses) or the non negligible two-factor interactions. This article provides for each 2n−(n−k) design: (1) the defining relation or the alias structure; (2) the k independent generators for sequencing the 2n−(n−k) runs by the generalized foldover scheme; and (3) the minimum cost represented by the total number of factor level changes between the 2n−(n−k) runs. All k main effects of the 2k experiment are excluded from the selection assignment process due to their nonlinear time trend resistance as well as excluding a total of (2k−1 –k +1) interactions violating the resolution IV requirement.

Minimum cost linear and quadratic trend free 2<SUP align="right">n−(n−k)</SUP> designs

This paper extends the degree of protection of factor effects in systematic 2n–(n–k) designs from the linear time trend into the quadratic and minimises the total number of factor level changes between runs (i.e., the cost) by constructing a catalogue of (2k–1 – k) minimum cost linear and quadratic trend free 2n–(n–k) designs (of resolution at least III) from the full 2k factorial experiment using the interactions-main effects assignment technique, where (2k–1 – (k – 1) – (k – 1)(k – 2) / 2) ≤ n ≤ (2k – 1 – k – k(k – 1) / 2) and k ≥ 4. All k main effects and their k(k – 1) / 2 two-factor interactions are excluded from the selection assignment process due to their non-quadratic time trend resistance. The paper provides for each design in the catalogue the following: 1) the independent interaction effects of the defining relation so that the alias structure can easily be identified; 2) the two-factor interactions that are linear/quadratic trend free besides the linear and quadratic trend freeness of all n main effects; 3) the k independent generators needed for the generalised fold-over scheme of Coster and Cheng (1988) to sequence the 2n–(n–k) runs; 4) the total number of factor level changes between the 2n–(n–k) successive runs (i.e., the experimental cost).

Confounding structure of two-level nonregular factorial designs

In design theory, the alias structure of regular fractional factorial designs is elegantly described with group theory. However, this approach cannot be applied to nonregular designs directly. For an arbitrary nonregular design, a natural question is how to describe the confounding relations between its effects, is there any inner structure similar to regular designs? The aim of this article is to answer this basic question. Using coefficients of indicator function, confounding structure of nonregular fractional factorial designs is obtained as linear constrains on the values of effects. A method to estimate the sparse significant effects in an arbitrary nonregular design is given through an example.

Efficient Designs With Minimal Aliasing

For some experimenters, a disadvantage of the standard optimal design approach is that it does not consider explicitly the aliasing of specified model terms with terms that are potentially important but are not included in the model. For example, when constructing an optimal design for a first-order model, aliasing of main effects and interactions is not considered. This can lead to designs that are optimal for estimation of the primary effects of interest, yet have undesirable aliasing structures. In this article, we construct exact designs that minimize the squared norm of the alias matrix subject to constraints on design efficiency. We demonstrate use of the method for the construction of screening and response surface designs.

Quarter-fraction factorial designs constructed via quaternary codes

The research of developing a general methodology for the construction of good nonregular designs has been very active in the last decade. Recent research by Xu and Wong [Statist. Sinica 17 (2007) 1191–1213] suggested a new class of nonregular designs constructed from quaternary codes. This paper explores the properties and uses of quaternary codes toward the construction of quarter-fraction nonregular designs. Some theoretical results are obtained regarding the aliasing structure of such designs. Optimal designs are constructed under the maximum resolution, minimum aberration and maximum projectivity criteria. These designs often have larger generalized resolution and larger projectivity than regular designs of the same size. It is further shown that some of these designs have generalized minimum aberration and maximum projectivity among all possible designs.

On a heuristic analysis of highly fractionated 2 n factorial experiments

The paper deals with a method for the analysis of highly fractionated factorial designs proposed by Raghavarao and Altan (2003). We show that the method will find active factors with almost any set of random numbers. Once that an alias set is found active, Raghavarao and Altan (2003) claim that their method can resolve the alias structure of the design and identify which of several confounded effects is active. We show that their method cannot do that. The error in Raghavarao and Altan's (2003) arguments lies in the fact that they treat a set of highly dependent (sometimes even identical) F-statistics as if they were independent.

The Use of Plackett–Burman Designs to Construct Split-Plot Designs

When some factors are hard to change and others are relatively easier, split-plot experiments are often an economic alternative to fully randomized designs. Split-plot experiments, with their structure of subplot arrays imbedded within whole-plot arrays, have a tendency to become large, particularly in screening situations when many factors are considered. To alleviate this problem, we explore, for the case of two-level designs, various ways to use orthogonal arrays of the Plackett–Burman type to reduce the number of individual tests. General construction principles are outlined, and the resulting alias structure is derived and discussed.