A dynamic sampling methodology for plasma etch processes using Gaussian process regression

Plasma etch is a key process in modern semiconductor manufacturing facilities as it offers process simplification and yet greater dimensional tolerances compared to wet chemical etch technology. The main challenge of operating plasma etchers is to maintain a consistent etch rate spatially and temporally for a given wafer and for successive wafers processed in the same etch tool. Etch rate measurements require expensive metrology steps and therefore in general only limited sampling is performed. Furthermore, the results of measurements are not accessible in real-time, limiting the options for run-to-run control. This paper investigates a Virtual Metrology (VM) enabled Dynamic Sampling (DS) methodology as an alternative paradigm for balancing the need to reduce costly metrology with the need to measure more frequently and in a timely fashion to enable wafer-to-wafer control. Using a Gaussian Process Regression (GPR) VM model for etch rate estimation of a plasma etch process, the proposed dynamic sampling methodology is demonstrated and evaluated for a number of different predictive dynamic sampling rules. © 2013 IEEE.

MSC-clustering and forward stepwise regression for virtual metrology in highly correlated input spaces

Increasingly semiconductor manufacturers are exploring opportunities for virtual metrology (VM) enabled process monitoring and control as a means of reducing non-value added metrology and achieving ever more demanding wafer fabrication tolerances. However, developing robust, reliable and interpretable VM models can be very challenging due to the highly correlated input space often associated with the underpinning data sets. A particularly pertinent example is etch rate prediction of plasma etch processes from multichannel optical emission spectroscopy data. This paper proposes a novel input-clustering based forward stepwise regression methodology for VM model building in such highly correlated input spaces. Max Separation Clustering (MSC) is employed as a pre-processing step to identify a reduced srt of well-conditioned, representative variables that can then be used as inputs to state-of-the-art model building techniques such as Forward Selection Regression (FSR), Ridge regression, LASSO and Forward Selection Ridge Regression (FCRR). The methodology is validated on a benchmark semiconductor plasma etch dataset and the results obtained are compared with those achieved when the state-of-art approaches are applied directly to the data without the MSC pre-processing step. Significant performance improvements are observed when MSC is combined with FSR (13%) and FSRR (8.5%), but not with Ridge Regression (-1%) or LASSO (-32%). The optimal VM results are obtained using the MSC-FSR and MSC-FSRR generated models. © 2012 IEEE.

Efficient Marginal Likelihood Computation for Gaussian Process Regression

In a Bayesian learning setting, the posterior distribution of a predictive model arises from a trade-off between its prior distribution and the conditional likelihood of observed data. Such distribution functions usually rely on additional hyperparameters which need to be tuned in order to achieve optimum predictive performance; this operation can be efficiently performed in an Empirical Bayes fashion by maximizing the posterior marginal likelihood of the observed data. Since the score function of this optimization problem is in general characterized by the presence of local optima, it is necessary to resort to global optimization strategies, which require a large number of function evaluations. Given that the evaluation is usually computationally intensive and badly scaled with respect to the dataset size, the maximum number of observations that can be treated simultaneously is quite limited. In this paper, we consider the case of hyperparameter tuning in Gaussian process regression. A straightforward implementation of the posterior log-likelihood for this model requires O(N^3) operations for every iteration of the optimization procedure, where N is the number of examples in the input dataset. We derive a novel set of identities that allow, after an initial overhead of O(N^3), the evaluation of the score function, as well as the Jacobian and Hessian matrices, in O(N) operations. We prove how the proposed identities, that follow from the eigendecomposition of the kernel matrix, yield a reduction of several orders of magnitude in the computation time for the hyperparameter optimization problem. Notably, the proposed solution provides computational advantages even with respect to state of the art approximations that rely on sparse kernel matrices.

Associations between Achievement Goal Orientations and Academic Performance Among Students at a UK Pharmacy School

Objective. To ascertain goal orientations of pharmacy students and establish whether associations exist between academic performance, gender, or year of study. Methods. Goal orientations were assessed using a validated questionnaire. Respondents were categorized as high or low performers based on university grades. Associations and statistical significance were ascertained using parametric and nonparametric tests and linear regression, as appropriate. Results. A response rate of 60.7% was obtained. High performers were more likely to be female than male. The highest mean score was for mastery approach; the lowest for work avoidance. The mean score for work avoidance was significantly greater for low performers than for high performers and for males than for females. First-year students were most likely to have top scores in mastery and performance approaches. Conclusion. It is encouraging that the highest mean score was for mastery approach orientation, as goal orientation may play a role in academic performance of pharmacy students.

On Regression Methods for Virtual Metrology in Semiconductor Manufacturing

Virtual metrology (VM) aims to predict metrology values using sensor data from production equipment and physical metrology values of preceding samples. VM is a promising technology for the semiconductor manufacturing industry as it can reduce the frequency of in-line metrology operations and provide supportive information for other operations such as fault detection, predictive maintenance and run-to-run control. The prediction models for VM can be from a large variety of linear and nonlinear regression methods and the selection of a proper regression method for a specific VM problem is not straightforward, especially when the candidate predictor set is of high dimension, correlated and noisy. Using process data from a benchmark semiconductor manufacturing process, this paper evaluates the performance of four typical regression methods for VM: multiple linear regression (MLR), least absolute shrinkage and selection operator (LASSO), neural networks (NN) and Gaussian process regression (GPR). It is observed that GPR performs the best among the four methods and that, remarkably, the performance of linear regression approaches that of GPR as the subset of selected input variables is increased. The observed competitiveness of high-dimensional linear regression models, which does not hold true in general, is explained in the context of extreme learning machines and functional link neural networks.

Kernel-based local order estimation of nonlinear nonparametric systems

We consider the local order estimation of nonlinear autoregressive systems with exogenous inputs (NARX), which may have different local dimensions at different points. By minimizing the kernel-based local information criterion introduced in this paper, the strongly consistent estimates for the local orders of the NARX system at points of interest are obtained. The modification of the criterion and a simple procedure of searching the minimum of the criterion, are also discussed. The theoretical results derived here are tested by simulation examples.

Forward and backward least angle regression for nonlinear system identification

A forward and backward least angle regression (LAR) algorithm is proposed to construct the nonlinear autoregressive model with exogenous inputs (NARX) that is widely used to describe a large class of nonlinear dynamic systems. The main objective of this paper is to improve model sparsity and generalization performance of the original forward LAR algorithm. This is achieved by introducing a replacement scheme using an additional backward LAR stage. The backward stage replaces insignificant model terms selected by forward LAR with more significant ones, leading to an improved model in terms of the model compactness and performance. A numerical example to construct four types of NARX models, namely polynomials, radial basis function (RBF) networks, neuro fuzzy and wavelet networks, is presented to illustrate the effectiveness of the proposed technique in comparison with some popular methods.

Non-parametric bill of quantities estimation of concrete road bridges' superstructure: an artificial neural networks approach

Bridge construction responds to the need for environmentally friendly design of motorways and facilitates the passage through sensitive natural areas and the bypassing of urban areas. However, according to numerous research studies, bridge construction presents substantial budget overruns. Therefore, it is necessary early in the planning process for the decision makers to have reliable estimates of the final cost based on previously constructed projects. At the same time, the current European financial crisis reduces the available capital for investments and financial institutions are even less willing to finance transportation infrastructure. Consequently, it is even more necessary today to estimate the budget of high-cost construction projects -such as road bridges- with reasonable accuracy, in order for the state funds to be invested with lower risk and the projects to be designed with the highest possible efficiency. In this paper, a Bill-of-Quantities (BoQ) estimation tool for road bridges is developed in order to support the decisions made at the preliminary planning and design stages of highways. Specifically, a Feed-Forward Artificial Neural Network (ANN) with a hidden layer of 10 neurons is trained to predict the superstructure material quantities (concrete, pre-stressed steel and reinforcing steel) using the width of the deck, the adjusted length of span or cantilever and the type of the bridge as input variables. The training dataset includes actual data from 68 recently constructed concrete motorway bridges in Greece. According to the relevant metrics, the developed model captures very well the complex interrelations in the dataset and demonstrates strong generalisation capability. Furthermore, it outperforms the linear regression models developed for the same dataset. Therefore, the proposed cost estimation model stands as a useful and reliable tool for the construction industry as it enables planners to reach informed decisions for technical and economic planning of concrete bridge projects from their early implementation stages.

Supervised Aggregative Feature Extraction for Big Data Time Series Regression

In many applications, and especially those where batch processes are involved, a target scalar output of interest is often dependent on one or more time series of data. With the exponential growth in data logging in modern industries such time series are increasingly available for statistical modeling in soft sensing applications. In order to exploit time series data for predictive modelling, it is necessary to summarise the information they contain as a set of features to use as model regressors. Typically this is done in an unsupervised fashion using simple techniques such as computing statistical moments, principal components or wavelet decompositions, often leading to significant information loss and hence suboptimal predictive models. In this paper, a functional learning paradigm is exploited in a supervised fashion to derive continuous, smooth estimates of time series data (yielding aggregated local information), while simultaneously estimating a continuous shape function yielding optimal predictions. The proposed Supervised Aggregative Feature Extraction (SAFE) methodology can be extended to support nonlinear predictive models by embedding the functional learning framework in a Reproducing Kernel Hilbert Spaces setting. SAFE has a number of attractive features including closed form solution and the ability to explicitly incorporate first and second order derivative information. Using simulation studies and a practical semiconductor manufacturing case study we highlight the strengths of the new methodology with respect to standard unsupervised feature extraction approaches.

LD Score regression distinguishes confounding from polygenicity in genome-wide association studies

Both polygenicity (many small genetic effects) and confounding biases, such as cryptic relatedness and population stratification, can yield an inflated distribution of test statistics in genome-wide association studies (GWAS). However, current methods cannot distinguish between inflation from a true polygenic signal and bias. We have developed an approach, LD Score regression, that quantifies the contribution of each by examining the relationship between test statistics and linkage disequilibrium (LD). The LD Score regression intercept can be used to estimate a more powerful and accurate correction factor than genomic control. We find strong evidence that polygenicity accounts for the majority of the inflation in test statistics in many GWAS of large sample size.