Extracted Basis Functions Research Articles

Addressing multi-step inverse heat transfer problems via reduced order models in a cooling process for polymer pipes with sparse measurements

In modern polymer pipe extrusion processes with connected post-processing steps, the intermediate cooling and conditioning process is essential for the resulting product quality. In such cases, computationally efficient models are required for real-time process control and optimization. Unfortunately, heat transfer coefficients are rarely known for actual production processes, leading to an inverse heat transfer problem. Existing methods mainly rely on computationally intensive numerical models or on data intensive neural networks. Both are not suitable, if only sparse measurements are available and a fast execution time is required. In contrast to that, we propose to use a proper orthogonal decomposition and project the governing heat equation onto extracted basis functions via the Galerkin method. This leads to an efficient and accurate reduced order model, which can be used for parameter identification and subsequent process control. Regarding the identification, we address the ill-posed problem with well-known relative constraints, l2 regularization and decomposition of coefficients based on the Nusselt number. Simulation and experimental results show that the heat transfer coefficients are consistently identified and robust against initial parameter guesses despite sparse and noisy temperature measurements. Consequently, the proposed method can be efficiently applied in cooling and conditioning processes in polymer extrusion and related applications such as identifying the heat transfer coefficients and thermal resistance in pressurized surge lines of nuclear power plants and brick walls in melting furnaces respectively.

Extracting Communication, Ranging and Test Waveforms with Regularized Timing from the Chaotic Lorenz System

We present an algorithm for extracting basis functions from the chaotic Lorenz system along with timing and bit-sequence statistics. Previous work focused on modifying Lorenz waveforms and extracting the basis function of a single state variable. Importantly, these efforts initiated the development of solvable chaotic systems with simple matched filters, which are suitable for many spread spectrum applications. However, few solvable chaotic systems are known, and they are highly dependent upon an engineered basis function. Non-solvable, Lorenz signals are often used to test time-series prediction schemes and are also central to efforts to maximize spectral efficiency by joining radar and communication waveforms. Here, we provide extracted basis functions for all three Lorenz state variables, their timing statistics, and their bit-sequence statistics. Further, we outline a detailed algorithm suitable for the extraction of basis functions from many chaotic systems such as the Lorenz system. These results promote the search for engineered basis functions in solvable chaotic systems, provide tools for joining radar and communication waveforms, and give an algorithmic process for modifying chaotic Lorenz waveforms to quantify the performance of chaotic time-series forecasting methods. The results presented here provide engineered test signals compatible with quantitative analysis of predicted amplitudes and regular timing.

Function-on-Function Partial Quantile Regression

A function-on-function linear quantile regression model, where both the response and predictors consist of random curves, is proposed by extending the classical quantile regression setting into the functional data to characterize the entire conditional distribution of functional response. In this paper, a functional partial quantile regression approach, a quantile regression analog of the functional partial least squares regression, is proposed to estimate the function-on-function linear quantile regression model. A partial quantile covariance function is first used to extract the functional partial quantile regression basis functions. The extracted basis functions are then used to obtain the functional partial quantile regression components and estimate the final model. Although the functional random variables belong to an infinite-dimensional space, they are observed in a finite set of discrete-time points in practice. Thus, in our proposal, the functional forms of the discretely observed random variables are first constructed via a finite-dimensional basis function expansion method. The functional partial quantile regression constructed using the functional random variables is approximated via the partial quantile regression constructed using the basis expansion coefficients. The proposed method uses an iterative procedure to extract the partial quantile regression components. A Bayesian information criterion is used to determine the optimum number of retained components. The proposed functional partial quantile regression model allows for more than one functional predictor in the model. However, the true form of the proposed model is unspecified, as the relevant predictors for the model are unknown in practice. Thus, a forward variable selection procedure is used to determine the significant predictors for the proposed model. Moreover, a case-sampling-based bootstrap procedure is used to construct pointwise prediction intervals for the functional response. The predictive performance of the proposed method is evaluated using several Monte Carlo experiments under different data generation processes and error distributions. The finite-sample performance of the proposed method is compared with the functional partial least squares method. Through an empirical data example, air quality data are analyzed to demonstrate the effectiveness of the proposed method.

Time-Warped Sparse Non-negative Factorization for Functional Data Analysis

This article proposes a novel time-warped sparse non-negative factorization method for functional data analysis. The proposed method on the one hand guarantees the extracted basis functions and their coefficients to be positive and interpretable, and on the other hand is able to handle weakly correlated functions with different features. Furthermore, the method incorporates time warping into factorization and hence allows the extracted basis functions of different samples to have temporal deformations. An efficient framework of estimation algorithms is proposed based on a greedy variable selection approach. Numerical studies together with case studies on real-world data demonstrate the efficacy and applicability of the proposed methodology.

Data‐Driven Inference of Thermosphere Composition During Solar Minimum Conditions

AbstractMass density variations can deviate from the expected behavior caused by temperature due to changes in the composition. Such deviations can be especially significant during solar minimum conditions. Model‐data differences are typically resolved through temperature corrections while overlooking the role of errors in lower boundary composition. In this work, we use a data‐driven methodology to simultaneously estimate thermosphere composition and temperature contributions to model‐data differences. The methodology uses modal decomposition to extract high‐dimensional, reduced order basis functions for the covariance of the neutral thermospheric species and temperature. The extracted basis functions are combined with CHAllenging Minisatellite Payload and Gravity Recovery And Climate Experiment mass density measurements using a nonlinear least squares solver. We demonstrate the methodology using the Naval Research Laboratory's empirical Mass Spectrometer and Incoherent Scatter (MSIS) model to derive the high‐dimensional basis functions. We characterize and quantify the contribution of temperature and lower boundary effects with oxygen and helium since the two species have a direct impact on drag and orbit prediction through gas‐surface interactions and mass density. We analyze the month of December in 2008, based on the work of Thayer et al. (2012), and estimate that lower boundary composition errors contribute approximately 50% of the model‐data differences.

Representation Discovery using Harmonic Analysis

Representations are at the heart of artificial intelligence (AI). This book is devoted to the problem of representation discovery: how can an intelligent system construct representations from its experience? Representation discovery re-parameterizes the state space - prior to the application of information retrieval, machine learning, or optimization techniques - facilitating later inference processes by constructing new task-specific bases adapted to the state space geometry. This book presents a general approach to representation discovery using the framework of harmonic analysis, in particular Fourier and wavelet analysis. Biometric compression methods, the compact disc, the computerized axial tomography (CAT) scanner in medicine, JPEG compression, and spectral analysis of time-series data are among the many applications of classical Fourier and wavelet analysis. A central goal of this book is to show that these analytical tools can be generalized from their usual setting in (infinite-dimensional) Euclidean spaces to discrete (finite-dimensional) spaces typically studied in many subfields of AI. Generalizing harmonic analysis to discrete spaces poses many challenges: a discrete representation of the space must be adaptively acquired; basis functions are not pre-defined, but rather must be constructed. Algorithms for efficiently computing and representing bases require dealing with the curse of dimensionality. However, the benefits can outweigh the costs, since the extracted basis functions outperform parametric bases as they often reflect the irregular shape of a particular state space. Case studies from computer graphics, information retrieval, machine learning, and state space planning are used to illustrate the benefits of the proposed framework, and the challenges that remain to be addressed. Representation discovery is an actively developing field, and the author hopes this book will encourage other researchers to explore this exciting area of research. Table of Contents: Overview / Vector Spaces / Fourier Bases on Graphs / Multiscale Bases on Graphs / Scaling to Large Spaces / Case Study: State-Space Planning / Case Study: Computer Graphics / Case Study: Natural Language / Future Directions