Optimal Approximate Designs Research Articles

Optimizing two‐variable gamma accelerated degradation tests with a semi‐analytical approach

AbstractIn this paper, we provide a semi‐analytical approach to determine the optimal designs for two‐variable gamma accelerated degradation tests under three criteria: ‐optimality, ‐optimality, and ‐optimality. We first use general equivalence theorem to prove that the optimal approximate designs only allocate test units at the four vertices of a rectangular design region, and the corresponding optimal proportion of total number of measurements at each stress level is derived. Next, we apply the concept of prescribed accuracy level and total experimental cost to further determine optimal integer designs. More specifically, a numerical approach is used to resolve the number of test units and number of measurements at each stress level. Finally, an example is presented to illustrate the proposed procedure and compare the efficiency of optimal designs under three optimal criteria.

MARADIV: Library of MAGIC-Based Approximate Restoring Array Divider Benchmark Circuits for In-Memory Computing Using Memristors

Among various arithmetic circuits, dividers are one of the costliest in terms of area, energy, and delay. Hence, approximate divider designs have been implemented to reduce their cost for error-resilient applications. However, approximate dividers based on memristors for in-memory computing (IMC) have not received any attention. In this work, we propose a library of 16 by 8 approximate restoring array dividers suitable for memristor-based IMC that can be used as benchmark circuits. We developed these benchmark circuits by performing an exhaustive design space exploration based on functional approximation. We used the state-of-the-art mapping tool, SIMPLER, to perform the mapping of the divider design to the memristor crossbar. We identified the pareto optimal approximate divider designs based on the MAGIC design style with varying output qualities. We compare our pareto optimal designs with the existing state-of-the-art CMOS-based approximate divider designs when mapped to IMCs. We also performed a case study on two image processing applications. Overall we propose MARADIV, a library of MAGIC design style-based approximate restoring array divider benchmark circuits tailored for memristor-based IMC. We believe that MARADIV’s benchmark designs can be used for future research in this direction. Hence, all pareto optimal approximate restoring array divider designs will be made available at https://github.com/agra-uni-bremen/tcasii2022-maradiv-lib.

Circular designs for total effects under interference models

This paper studies circular designs for interference models, where a treatment assigned to a plot also affects its neighboring plots within a block. For the purpose of estimating total effects, the circular neighbor balanced design was shown to be universally optimal among designs which do not allow treatments to be neighbors of themselves. Our study shows that self-neighboring block sequences are actually the main ingredient for an optimal design. Here, we adopt the approximate design framework and study optimal designs in the whole design space. Our approach is flexible enough to accommodate all possible design parameters, that is the block size and the number of blocks and treatments. This approach can be broken down into two main steps: the identification of the minimal supporting set of block sequences and the optimality condition built on it. The former is critical for reducing the computational time from almost infinity to seconds. Meanwhile, the task of finding the minimal set is normally achieved through numerical methods, which can only handle small block sizes. Our approach is of a hybrid nature in order to deal with all design sizes. When block size is not large, we provide explicit expressions of the minimal set instead of relying on numerical methods. For larger block sizes when a typical numerical method would fail, we theoretically derived a reasonable size intermediate set of sequences, from which the minimal set can be quickly derived through a customized algorithm. Taking it further, the optimality conditions allow us to obtain both symmetric and asymmetric designs. Lastly, we also investigate the trade-off issue between circular and noncircular designs, and provide guidelines on the choices.

Practical implementation of optimal experimental design using the fractional-order Fricke–Morse bioimpedance model

This paper provides, for the first time, the application of the Optimal Experimental Design (OED) theory. Two algorithms for computing exact and approximate optimal designs have been adapted for the fractional-order Fricke–Morse circuit model (which is widely used to describe experimental bioimpedance data). Frequencies at which the impedance is measured are optimized, while reducing the measurement acquisition time and maximizing the information about the fractional-order electrical behaviour of the biological tissue. As a practical implementation of this methodology, for a sample of apple tissue, D-optimal approximate and exact designs are computed to obtain the best estimates of the parameters values according to a criteria. These designs were compared with the classical design commonly used by practitioners showing the efficiencies of the optimal designs. The application of OED theory to this type of problems opens up many possibilities for future research.

Efficient Approximate Multiplier Design Based on Hybrid Higher Radix Booth Encoding

Partial products of higher radix are approximated by partial product segmentation. The encoder skips few lower order bits to segment the partial products for approximation. For a particular type of approximation, one can have many options of choosing the higher radix for hybrid encoding. However, the selection of higher radix is the key to obtain an optimal approximate hybrid multiplier design. In this paper, we made an in-depth study on radix-64 and radix-256 encoding for 4-bit and 5-bit approximations. Proposed a two-term radix-64 encoding and three-term radix-256 encoding for 4-bit approximation to calculate partial-products with lowest maximum absolute error (MAE). Further, we have proposed a simple rule to select the higher radix for hybrid encoding to obtain an efficient approximate multiplier design. To demonstrate the usefulness of the selection rule, two hybrid encoding schemes named R4R64 and R4R256 are formulated by combining radix-4 separately with proposed two-term radix-64 and three-term radix-256 encoding schemes. Separate multiplier designs are derived using the proposed hybrid encoding. Comparison result shows that both the proposed approximate multipliers have identical output accuracy, but the multiplier design based on proposed hybrid R4R64 encoding is more hardware efficient than the other. Compared with the existing approximate multiplier based on hybrid R4R256 encoding for word-length 12, the proposed multiplier design based on R4R64 hybrid encoding involves nearly 15% less area-delay product (ADP) and a marginal 4% less power-delay-product (PDP), but calculates output with higher accuracy. The 2-D DCT structure using proposed R4R64 multiplier involves 13% lesser ADP and 14% less energy consumption when compared with the 2-D DCT structure based on existing hybrid R4R256 multiplier design and computes reconstructed images with nearly 10dB higher pick signal-to-noise ratio (PSNR).

Blockchain Operations in the Presence of Security Concerns

Problem definition: A blockchain payment system, such as Bitcoin or Ethereum, validates electronic transactions and stores them in a chain of blocks without a central authority. Miners with computing power compete for the rights to create blocks according to a preset protocol, referred to as hashing or mining, and, in return, earn fees paid by users who submit transactions. Because of security concerns caused by decentralization, a transaction is confirmed after a number of additional blocks are subsequently extended to the block containing it. This confirmation latency introduces an intricate interplay between miners and users. This paper provides approximate system equilibria and studies optimal designs of a blockchain. Methodology/results: The hashing process is essentially a single-server queue with batch services based on a fee-based priority discipline, and confirmation latency adds complexity to the equilibrium behavior and optimal design. We analyze how miners’ participation decisions interact with users’ participation and fee decisions and identify optimal designs when the goal is to maximize the throughput or social welfare. We validate our model and conduct numerical studies using data from Bitcoin. Managerial implications: By incorporating security issues, we uncover the interdependence of the decisions between users and miners and the driver for nonzero entrance fees in practice. We show that miners and users may end up in either a vicious or virtuous cycle, depending on the initial system state. By allowing the entrance fee to be a design parameter, we are able to establish that it is optimal to simply run a blockchain system at its full capacity and a block size as small as possible. Funding: This work was supported by the Hong Kong Research Grants Council [Grants 16200019, 16200617, 16200821, 16208120, and 16214121]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/msom.2023.1197 .

Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming

The systematic design of exact optimal designs of experiments is typically challenging, as it results in nonconvex optimization problems. The literature on the computation of model-based exact optimal designs of experiments via mathematical programming, when the covariates are categorical variables, is still scarce. We propose mixed-integer semidefinite programming formulations, to find exact D-, A- and I-optimal designs for linear models, and locally optimal designs for nonlinear models when the design domain is a finite set of points. The strategy requires: (i) the generation of a set of candidate treatments; (ii) the formulation of the optimal design problem as a mixed-integer semidefinite program; and (iii) its solution, employing appropriate solvers. For comparison, we use semidefinite programming-based formulations to find equivalent approximate optimal designs. We demonstrate the application of the algorithm with various models, considering both unconstrained and constrained setups. Equivalent approximate optimal designs are used for comparison.

Analytical method for suboptimal design of dynamic absorber for parametrically excited system

Parametric excitation is induced due to systemically unstable equilibrium states and often results in undesired vibration in mechanical systems. Many researchers have shown that the use of a dynamic absorber with an appropriate natural frequency and damping ratio is effective for suppressing parametric excitation. However, unlike the situation in a forced vibration system, it is difficult to find analytical expressions to determine the optimal design of the dynamic absorber for a parametrically excited system, which inevitably makes the design inefficient. This study developed a method for obtaining analytical suboptimal designs, specifically approximate optimal designs, for a dynamic absorber for parametric excitation through two kinds of free vibration systems constructed by approximating the original system. A comparison between the results for the suboptimal designs and an accurate optimal design demonstrated that an effective suboptimal design is achieved using a free vibration system that has an appropriate negative damping term derived from the instability of the original parametrically excited system. The developed approach is expected to provide an efficient option for suppressing parametric excitation.

Optimal Approximate Choice Designs for a Two-step Coffee Choice, Taste and Choice Again Experiment

AbstractThis work deals with consumers' preferences about coffee. Firstly, a choice experiment is performed on a sample of potential consumers. Following this, a sensory test involving the tasting of two varieties of coffee is carried out with the respondents, after which the same choice experiment is supplied to them again. An innovative approach for building heterogeneous choice designs is specifically developed for the case-study, based on approximate design theory and compound design criterion. Panel Mixed Logit models are used, thereby allowing for the inclusion of correlation among consumers' responses; choice-sets are supplied to a proportion of respondents according to optimal weights. The estimation results of the Panel Mixed Logit model are satisfactory, confirming the validity of the proposed approach.

Approximate and Exact Optimal Designs for Paired Comparison Experiments

In this article, the problem of finding optimal paired comparison approximate and exact designs for the identification of main effects and two and three and four attribute interactions, when the alternatives are characterized by either full profiles or partial profiles, is considered. The resulting designs are also optimal under the indifference assumption of equal choice probabilities for a multinomial logit model when the choice sets are pairs. AMS subject classification: 62K05, 62J15, 62K15

Equivalence theorems for c and DA‐optimality for linear mixed effects models with applications to multitreatment group assignments in health care

AbstractWe construct and ‐optimal approximate designs for linear mixed models with group‐specific treatment for estimating parameters or contrasts in the population parameters. We establish equivalence theorems to confirm optimality of these designs under a linear mixed model and provide illustrative application to find , and ‐optimal designs for polynomial and fractional polynomial models with multitreatment group assignments. For more complex models, we briefly review metaheuristics and their potential applications to find various optimal designs, including optimal designs for problems considered here and their extensions.

Efficient computational algorithms for approximate optimal designs

In this paper, we propose two simple yet efficient computational algorithms to obtain approximate optimal designs for multi-dimensional linear regression on a large variety of design spaces. We focus on the two commonly used optimal criteria, D- and A-optimal criteria. For D-optimality, we provide an alternative proof for the monotonic convergence for D-optimal criterion and propose an efficient computational algorithm to obtain the approximate D-optimal design. We further show that the proposed algorithm converges to the D-optimal design and then proves that the approximate D-optimal design converges to the continuous D-optimal design under certain conditions. For A-optimality, we provide an efficient algorithm to obtain approximate A-optimal design and conjecture the monotonicity of the proposed algorithm. Numerical comparisons suggest that the proposed algorithms perform well and they are comparable or superior to some existing algorithms.

Optimal design of multifactor experiments via grid exploration

We propose an algorithm for computing efficient approximate experimental designs that can be applied in the case of very large grid-like design spaces. Such a design space typically corresponds to the set of all combinations of multiple genuinely discrete factors or densely discretized continuous factors. The proposed algorithm alternates between two key steps: (1) the construction of exploration sets composed of star-shaped components and separate, highly informative design points and (2) the application of a conventional method for computing optimal approximate designs on medium-sized design spaces. For a given design, the star-shaped components are constructed by selecting all points that differ in at most one coordinate from some support point of the design. Because of the reliance on these star sets, we call our algorithm the galaxy exploration method (GEX). We demonstrate that GEX significantly outperforms several state-of-the-art algorithms when applied to D-optimal design problems for linear, generalized linear and nonlinear regression models with continuous and mixed factors. Importantly, we provide a free R code that permits direct verification of the numerical results and allows researchers to easily compute optimal or nearly optimal experimental designs for their own statistical models.

On efficient exact experimental designs for ordered treatments

In a recent paper Singh and Davidov (2019) derive approximate optimal designs for experiments with ordered treatments. Specifically, maxi–min and intersection–union designs were explored. These designs, which address different types of hypothesis testing problems, provide a substantial improvement over standard designs in terms of power, or equivalently, sample size requirements. In practice however, exact, not approximate designs are used. Therefore, in this paper, we develop methods for finding efficient exact designs for the testing problems considered in Singh and Davidov (2019). The proposed designs are compared numerically to some well known existing designs and it is shown that the new designs require fewer experimental units to attain prespecified power. A thorough sensitivity analysis shows that the proposed designs are robust against possible misspecification of the parameters under the alternative and the order relation among the treatment groups.

Optimal experimental design for linear time invariant state–space models

The linear time invariant state–space model representation is common to systems from several areas ranging from engineering to biochemistry. We address the problem of systematic optimal experimental design for this class of model. We consider two distinct scenarios: (i) steady-state model representations and (ii) dynamic models described by discrete-time representations. We use our approach to construct locally D-optimal designs by incorporating the calculation of the determinant of the Fisher Information Matrix and the parametric sensitivity computation in a Nonlinear Programming formulation. A global optimization solver handles the resulting numerical problem. The Fisher Information Matrix at convergence is used to determine model identifiability. We apply the methodology proposed to find approximate and exact optimal experimental designs for static and dynamic experiments for models representing a biochemical reaction network where the experimental purpose is to estimate kinetic constants.

Optimal experimental designs for treatment contrasts in heteroscedastic models with covariates

In clinical trials, the response of a given subject often depends on the selected treatment as well as on some covariates. We study optimal approximate designs of experiments in the models with treatment and covariate effects. We allow for the variances of the responses to depend on the chosen treatments, which introduces heteroscedasticity into the models. For estimating systems of treatment contrasts and linear functions of the covariates, we extend known results on D-optimality of product designs by providing product designs that are optimal with respect to general eigenvalue-based criteria. In particular, A- and E-optimal product designs are obtained. We then formulate a method based on linear programming for constructing optimal designs with smaller supports from the optimal product designs. The sparser designs can be more easily converted to practically applicable exact designs. The provided results and the proposed sparsification method are demonstrated in some examples.

Ascent with quadratic assistance for the construction of exact experimental designs

In the area of statistical planning, there is a large body of theoretical knowledge and computational experience concerning so-called optimal approximate designs of experiments. However, for an approximate design to be realizable, it must be converted into an exact, i.e., integer, design, which is usually done via rounding procedures. Although rapid, rounding procedures often yield worse exact designs than heuristics that do not require approximate designs at all. In this paper, we build on an alternative principle of utilizing optimal approximate designs for the computation of optimal, or nearly-optimal, exact designs. The principle, which we call ascent with quadratic assistance (AQuA), is an integer programming method based on the quadratic approximation of the design criterion in the neighborhood of the optimal approximate information matrix. To this end, we present quadratic approximations of all Kiefer’s criteria with an integer parameter, including D- and A-optimality and, by a model transformation, I-optimality. Importantly, we prove a low-rank property of the associated quadratic forms, which enables us to use AQuA efficiently and apply it to large design spaces. We numerically demonstrate the robustness and superior performance of the proposed method for selected statistical models under various types of experimental constraints. We also show how can iterative application of AQuA be used for a stratified information-based subsampling of large datasets under a lower bound on the quality and an upper bound on the cost of the subsample.

A comparison of general-purpose optimization algorithms for finding optimal approximate experimental designs

Several common general purpose optimization algorithms are compared for finding A- and D-optimal designs for different types of statistical models of varying complexity, including high dimensional models with five and more factors. The algorithms of interest include exact methods, such as the interior point method, the Nelder–Mead method, the active set method, the sequential quadratic programming, and metaheuristic algorithms, such as particle swarm optimization, simulated annealing and genetic algorithms. Several simulations are performed, which provide general recommendations on the utility and performance of each method, including hybridized versions of metaheuristic algorithms for finding optimal experimental designs. A key result is that general-purpose optimization algorithms, both exact methods and metaheuristic algorithms, perform well for finding optimal approximate experimental designs.

CVX‐based algorithms for constructing various optimal regression designs

AbstractCVX‐based numerical algorithms are widely and freely available for solving convex optimization problems but their applications to solve optimal design problems are limited. Using the CVX programs in MATLAB, we demonstrate their utility and flexibility over traditional algorithms in statistics for finding different types of optimal approximate designs under a convex criterion for nonlinear models. They are generally fast and easy to implement for any model and any convex optimality criterion. We derive theoretical properties of the algorithms and use them to generate new A‐, c‐, D‐ and E‐optimal designs for various nonlinear models, including multi‐stage and multi‐objective optimal designs. We report properties of the optimal designs and provide sample CVX program codes for some of our examples that users can amend to find tailored optimal designs for their problems. The Canadian Journal of Statistics 47: 374–391; 2019 © 2019 Statistical Society of Canada

A Randomized Exchange Algorithm for Computing Optimal Approximate Designs of Experiments

We propose a class of subspace ascent methods for computing optimal approximate designs that covers existing algorithms as well as new and more efficient ones. Within this class of methods, we construct a simple, randomized exchange algorithm (REX). Numerical comparisons suggest that the performance of REX is comparable or superior to that of state-of-the-art methods across a broad range of problem structures and sizes. We focus on the most commonly used criterion of D-optimality, which also has applications beyond experimental design, such as the construction of the minimum-volume ellipsoid containing a given set of data points. For D-optimality, we prove that the proposed algorithm converges to the optimum. We also provide formulas for the optimal exchange of weights in the case of the criterion of A-optimality, which enable one to use REX and some other algorithms for computing A-optimal and I-optimal designs. Supplementary materials for this article are available online.