Real-world problems often consist of multiple conflicting objectives to be optimized simultaneously, featuring a set of Pareto-optimal solutions. Estimating the entire Pareto front can be computationally expensive, and is not always necessary, as decision makers (DMs) will likely be interested only in specific regions of the Pareto front. In the absence of knowledge about the DM preferences, the so-called knees in the Pareto front are considered to be particularly attractive. In this article, we propose using Thompson sampling in the Bayesian optimization framework to estimate the location of the knee regions in a data-efficient manner. Our experimental results show that the proposed methods accurately locate the knee regions after a very small number of evaluations, providing a computationally efficient approach to single- and multiknee detection in multiobjective optimization.

Predicting the disability progression in multiple sclerosis remains a significant challenge. Statistical features extracted from evoked potential signals are often discussed in literature as interesting biomarkers, yet their practical adoption remains limited due to their low effectiveness. Conversely, the use of deep neural network feature encoders, which allow for extracting more expressive biomarkers, is hindered by the limited availability of data. This study proposes a novel method of enhancing statistically extracted features by aligning hospital visits to embed the progression of the disease within the features. The results demonstrate a significant improvement in predicting disability progression using the statistically extracted features. Insight is provided through a study of correlation and importance of the enhanced features.

We present Parallel Feasible Pareto Frontier Entropy Search ($\{\mathrm{PF}\}^2$ES) — a novel information-theoretic acquisition function for multi-objective Bayesian optimization supporting unknown constraints and batch queries. Due to the complexity of characterizing the mutual information between candidate evaluations and (feasible) Pareto frontiers, existing approaches must either employ crude approximations that significantly hamper their performance or rely on expensive inference schemes that substantially increase the optimization’s computational overhead. By instead using a variational lower bound, $\{\mathrm{PF}\}^2$ES provides a low-cost and accurate estimate of the mutual information. We benchmark $\{\mathrm{PF}\}^2$ES against other information-theoretic acquisition functions, demonstrating its competitive performance for optimization across synthetic and real-world design problems.

As visible light positioning combines illumination and indoor localization, the relationship between the uniformity index as a quality measurement of the illumination and the precision of the positioning technique is investigated. It is demonstrated that the classic multilateration approach leads to a degradation of the accuracy as the uniformity improves. When a Gaussian Process is deployed, the accuracy improves for better uniformity indices. Further normalization of the area size leads to a generic curve that can be used to estimate the required training set size when a Gaussian Process is used given the required accuracy and observation plane size.

Open-set recognition (OSR) has achieved significant importance in recent years. For a robust recognition system, we need to identify the right class from a myriad of knowns and unknowns. In this work, we build and compare OSR systems for patient activity recognition (PAR) using compact radar sensors in a hospital setting. Radar sensors are an important part of a privacy-preserving monitoring system. Specifically, the proposed approach is based on a deep discriminative representation network (DDRN) trained using the large margin cosine loss (LMCL) and triplet loss (TL). A probability of an inclusion model in the embedding space based on the Weibull distribution is able to separate knowns from unknowns. This overall approach limits the risk of open space and enables us to easily identify any unknown activities. Our experiments show that the proposed approach is significantly better for open-set human activity recognition (HAR) with radar when compared with the state-of-the-art open-set approaches.

We study the inference of mean-variance robustness measures to quantify input uncertainty under the Gaussian Process (GP) framework. These measures are widely used in applications where the robustness of the solution is of interest, for example, in engineering design. While the variance is commonly used to characterize the robustness, Bayesian inference of the variance using GPs is known to be challenging. In this paper, we propose a Spectral Representation of Robustness Measures based on the GP’s spectral representation, i.e., an analytical approach to approximately infer both robustness measures for normal and uniform input uncertainty distributions. We present two approximations based on different Fourier features and compare their accuracy numerically. To demonstrate their utility and efficacy in robust Bayesian Optimization, we integrate the analytical robustness measures in three standard acquisition functions for various robust optimization formulations. We show their competitive performance on numerical benchmarks and real-life applications.

This work develops and benchmarks surrogate models for Dynamic Phasor (DP) simulation of electrical drives. DP simulations of complex systems may be time-consuming due to the increased number of equations. Thus, it is desirable to have a data-driven approach to compute the critical state/control variables and power losses. The surrogate models are intended to be used as a steady-state equivalent of the DP simulation model. We consider the Gaussian Process (GP), Multi Layer Perceptron, and Random Forest as surrogate models. Among other techniques, GPs are found to have good accuracy. Moreover, GPs are data-efficient and have desirable properties, such as built-in uncertainty quantification. The study shows that the GP performs better compared to other techniques in terms of the Mean Squared Error of the prediction, while still being very fast to evaluate. We illustrate the potential of these surrogate models to also predict transient behavior.

Sleep apnea is one of the most common sleep-related breathing disorders. It is diagnosed through an overnight sleep study in a specialized sleep clinic. This setup is expensive and the number of beds and staff are limited, leading to a long waiting time. To enable more patients to be tested, and repeated monitoring for diagnosed patients, portable sleep monitoring devices are being developed. These devices automatically detect sleep apnea events in one or more respiration-related signals. There are multiple methods to measure respiration, with varying levels of signal quality and comfort for the patient. In this study, the potential of using the bio-impedance (bioZ) of the chest as a respiratory surrogate is analyzed. A novel portable device is presented, combined with a two-phase Long Short-Term Memory (LSTM) deep learning algorithm for automated event detection. The setup is benchmarked using simultaneous recordings of the device and the traditional polysomnography in 25 patients. The results demonstrate that using only the bioZ, an area under the precision-recall curve of 46.9% can be achieved, which is on par with automatic scoring using a polysomnography respiration channel. The sensitivity, specificity and accuracy are 58.4%, 76.2% and 72.8% respectively. This confirms the potential of using the bioZ device and deep learning algorithm for automatically detecting sleep respiration events during the night, in a portable and comfortable setup.

Electrocardiogram signals are often used in medicine. An important aspect of analyzing this data is identifying and classifying the type of beat. This classification is often done through an automated algorithm. Recent advancements in neural networks and deep learning have led to high classification accuracy. However, adoption of neural network models into clinical practice is limited due to the black-box nature of the classification method. In this work, the use of variational auto encoders to learn human-interpretable encodings for the beat types is analyzed. It is demonstrated that using this method, an interpretable and explainable representation of normal and paced beats can be achieved with neural networks.

Often in the design process of an engineer, the design specifications of the system are not completely known initially. However, usually there are some physical constraints which are already known, corresponding to a region of interest in the design space that is called feasible. These constraints often have no analytical form but need to be characterised based on expensive simulations or measurements. Therefore, it is important that the feasible region can be modeled sufficiently accurate using only a limited amount of samples. This can be solved by using active learning techniques that minimize the amount of samples w.r.t. what we try to model. Most active learning strategies focus on classification models or regression models with classification accuracy and regression accuracy in mind respectively. In this work, regression models of the constraints are used, but only the (in) feasibility is of interest. To tackle this problem, an information-theoretic sampling strategy is constructed to discover these regions. The proposed method is then tested on two synthetic examples and one engineering example and proves to outperform the current state-of-the-art.

Sleep apnea is one of the most common sleep disorders and the consequences of undiagnosed sleep apnea can be very severe, ranging from increased blood pressure to heart failure. However, many people are often unaware of their condition. The gold standard for diagnosing sleep apnea is an overnight polysomnography in a dedicated sleep laboratory. Yet, these tests are expensive and beds are limited as trained staff needs to analyze the entire recording. An automated detection method would allow a faster diagnosis and more patients to be analyzed. Most algorithms for automated sleep apnea detection use a set of human-engineered features, potentially missing important sleep apnea markers. In this paper, we present an algorithm based on state-of-the-art deep learning models for automatically extracting features and detecting sleep apnea events in respiratory signals. The algorithm is evaluated on the Sleep-Heart-Health-Study-1 dataset and provides per-epoch sensitivity and specificity scores comparable to the state of the art. Furthermore, when these predictions are mapped to the apnea-hypopnea index, a considerable improvement in per-patient scoring is achieved over conventional methods. This paper presents a powerful aid for trained staff to quickly diagnose sleep apnea.

Non-intrusive load monitoring detects active appliances in a household (and their power consumption) from measuring the aggregated power at just one point in that household. Our previous works focused on classifying a single appliance, assuming that the voltage and current trace could be isolated from an aggregated signal by considering the difference in current before and after the event. In this paper, we show that this assumption holds and that it is a viable approach in practice. We experimentally validate this for two classification methods we proposed earlier: (1) random forests using elliptical Fourier descriptors of the appliances' VI trajectories and (2) convolutional neural networks using the appliances' VI images. We benchmark these approaches on the aggregated data from the 2018 version of PLAID. We obtain, respectively for each of these classifiers, a maximal Fmacro-measure of 85.31% and 87.95%. We also show that using submetered data for training does not improve the performance.

Contemporary surveillance systems mainly use video cameras as their primary sensor. However, video cameras possess fundamental deficiencies, such as the inability to handle low-light environments, poor weather conditions, and concealing clothing. In contrast, radar devices are able to sense in pitch-dark environments and to see through walls. In this paper, we investigate the use of micro-Doppler (MD) signatures retrieved from a low-power radar device to identify a set of persons based on their gait characteristics. To that end, we propose a robust feature learning approach based on deep convolutional neural networks. Given that we aim at providing a solution for a real-world problem, people are allowed to walk around freely in two different rooms. In this setting, the IDentification with Radar data data set is constructed and published, consisting of 150 min of annotated MD data equally spread over five targets. Through experiments, we investigate the effectiveness of both the Doppler and time dimension, showing that our approach achieves a classification error rate of 24.70% on the validation set and 21.54% on the test set for the five targets used. When experimenting with larger time windows, we are able to further lower the error rate.

Constructing surrogate models of high-dimensional complex black-box systems from simulation-based data requires an appropriate choice of surrogate model type, as well as identification of the most influential input parameters. As including irrelevant input parameters results in a longer surrogate model training process and potentially increases the risk of overfitting, it is important to identify a small set of relevant parameters during the adaptive modeling phase of the surrogate modeling process. A multi-objective optimization step is proposed to identify both the appropriate model type as well as a parameters subset. The obtained model can be used for evaluation intensive applications such as exploration, sensitivity analysis or optimization.

High Dimensional Model Representation (HDMR) offers efficient ways to approximate computation-intensive high- dimensional black-box functions. The distinctive nature of HDMR allows a high-dimensional problem to be decomposed into a low-dimensional function or a combination of various low-dimensional functions, thus making it more attractive than other popular metamodelling approaches such as Kriging, Radial basis function, etc. However, the computational cost of HDMR is still a bottleneck for high-dimensional problems. In this work, a hybrid sequential sampling based Kriging metamodelling technique is integrated with HDMR to improve the computational efficiency of HDMR for high-dimensional problems. The performance of the proposed metamodelling approach for high-dimensional problems is validated with various benchmark mathematical problems of a wide scope of dimensionalities.

BACKGROUND: Many algorithms have been developed to infer the topology of gene regulatory networks from gene expression data. These methods typically produce a ranking of links between genes with associated confidence scores, after which a certain threshold is chosen to produce the inferred topology. However, the structural properties of the predicted network do not resemble those typical for a gene regulatory network, as most algorithms only take into account connections found in the data and do not include known graph properties in their inference process. This lowers the prediction accuracy of these methods, limiting their usability in practice. RESULTS: We propose a post-processing algorithm which is applicable to any confidence ranking of regulatory interactions obtained from a network inference method which can use, inter alia, graphlets and several graph-invariant properties to re-rank the links into a more accurate prediction. To demonstrate the potential of our approach, we re-rank predictions of six different state-of-the-art algorithms using three simple network properties as optimization criteria and show that Netter can improve the predictions made on both artificially generated data as well as the DREAM4 and DREAM5 benchmarks. Additionally, the DREAM5 E.coli. community prediction inferred from real expression data is further improved. Furthermore, Netter compares favorably to other post-processing algorithms and is not restricted to correlation-like predictions. Lastly, we demonstrate that the performance increase is robust for a wide range of parameter settings. Netter is available at http://bioinformatics.intec.ugent.be. CONCLUSIONS: Network inference from high-throughput data is a long-standing challenge. In this work, we present Netter, which can further refine network predictions based on a set of user-defined graph properties. Netter is a flexible system which can be applied in unison with any method producing a ranking from omics data. It can be tailored to specific prior knowledge by expert users but can also be applied in general uses cases. Concluding, we believe that Netter is an interesting second step in the network inference process to further increase the quality of prediction.

Wireless systems often need to optimize multiple conflicting objectives (low delay, high reliability, and low cost), which are difficult to fulfill simultaneously. In such cases, the wireless system exhibits multiple optimal operation points, referred to as the optimal Pareto front (OPF). However, due to the large number of parameter settings to be evaluated and the time-consuming nature of performing wireless experiments, it is typically not possible to identify the OPF by exhaustively evaluating all possible settings. Instead, for many use cases, an approximation is good enough. To this end, this paper applies a multi-objective surrogate-based optimization (MOSBO) toolbox to efficiently optimize wireless systems and approximate the OPF using a limited number of iterations. Moreover, a real Wi-Fi conferencing scenario is optimized that has two conflicting objectives (exposure and audio quality) and four configurable parameters (Tx-Power, Tx-Rate, Codec Bit-Rate, and Codec Frame-Length). The benefits of using the MOSBO approach for such a network problem is demonstrated by approximating the OPF using 94 iterations instead of requiring the exploration of 6528 different parameter combinations, while still dominating 96.58% of the complete design space.

A large amount of research focuses on experimentally optimizing the performance of wireless solutions. Finding the optimal performance settings typically requires investigating all possible combinations of design parameters, while the number of required experiments increases exponentially for each considered design parameter. The aim of this paper is to analyze the applicability of global optimization techniques to reduce the optimization time of wireless experimentation. In particular, the paper applies the Efficient Global Optimization (EGO) algorithm implemented in the SUrrogate MOdeling (SUMO) toolbox inside a wireless testbed. Moreover, to cope with the unpredictable nature of wireless testbeds, the paper applies an experiment outlier detection which monitors outside interference and verifies the validity of conducted experiments. The proposed techniques are implemented and evaluated in a wireless testbed using a realistic wireless conferencing scenario. The performance gain and experimentation time of a SUMO optimized experiment is compared against an exhaustively searched experiment. In our proof of concept, it is shown that the proposed SUMO optimizer reaches 99.79% of the global optimum performance while requiring 8.67 times less experiments compared to the exhaustive search experiment.

Surrogate models or metamodels are widely used in the realm of engineering for design optimization to minimize the number of computationally expensive simulations. Most practical problems often have conflicting objectives, which lead to a number of competing solutions which form a Pareto front. Multi-objective surrogate-based constrained optimization algorithms have been proposed in literature, but handling constraints directly is a relatively new research area. Most algorithms proposed to directly deal with multi-objective optimization have been evolutionary algorithms (Multi-Objective Evolutionary Algorithms - MOEAs). MOEAs can handle large design spaces but require a large number of simulations, which might be infeasible in practice, especially if the constraints are expensive. A multi-objective constrained optimization algorithm is presented in this paper which makes use of Kriging models, in conjunction with multi-objective probability of improvement (PoI) and probability of feasibility (PoF) criteria to drive the sample selection process economically. The efficacy of the proposed algorithm is demonstrated on an analytical benchmark function, and the algorithm is then used to solve a microwave filter design optimization problem.

Reduced state-space models obtained by model order reduction methods must be accurate over the whole frequency range of interest and must also preserve passivity.In this paper, we propose multipoint reduction technique using reflective exploration for adaptively choosing the expansion points.The projection matrices obtained from the expansion points are merged to form the overall projection matrix.In order to obtain a more compact model the projection matrix is truncated based on its singular values.Finally, the reduced order model is obtained, while ensuring that the passivity of the reduced system is preserved during the reduction process.

This paper presents a technique for automatically extracting analytical behavioral models from the netlist of a nonlinear analog circuit. Subsequent snapshots of the internal circuit Jacobian are sampled during time-domain analysis and are then processed into Transfer Function Trajectories (TFT). The TFT data project the nonlinear dynamics of the system onto a hyperplane in the mixed state-space/frequency domain. Next Recursive Vector Fitting (RVF) algorithm is used to extract an analytical Hammerstein model out of the TFT data in an automated fashion. The resulting RVF model equations are implemented as an accurate nonlinear behavioral model in the time domain. The model is guaranteed stable by construction and can trade off complexity for accuracy. The technique is validated on a high-speed analog buffer circuit containing 70 linear and nonlinear components, showing a 7X speedup.

A novel state-space realization for parametric macromodeling is proposed in this paper.A judicious choice of the state-space realization is required to account for the generally assumed smoothness of the state-space matrices with respect to the design parameters.This is used in combination with suitable interpolation schemes to interpolate a set of state-space matrices, and hence the poles and residues indirectly, in order to build accurate parametric macromodels.The key points are the choice of a proper pivot matrix and the solution of a Sylvester equation for pole placement.Pertinent numerical examples validate the proposed state-space realization for parametric macromodeling.

In order to ensure optimal quality of experience toward end users during video streaming, automatic video quality assessment becomes an important field-of-interest to video service providers. Objective video quality metrics try to estimate perceived quality with high accuracy and in an automated manner. In traditional approaches, these metrics model the complex properties of the human visual system. More recently, however, it has been shown that machine learning approaches can also yield competitive results. In this paper, we present a novel no-reference bitstream-based objective video quality metric that is constructed by genetic programming-based symbolic regression. A key benefit of this approach is that it calculates reliable white-box models that allow us to determine the importance of the parameters. Additionally, these models can provide human insight into the underlying principles of subjective video quality assessment. Numerical results show that perceived quality can be modeled with high accuracy using only parameters extracted from the received video bitstream.

The use of Surrogate Based Optimization (SBO) is widely spread in engineering design to reduce the number of computational expensive simulations. However, “real-world” problems often consist of multiple, conflicting objectives leading to a set of equivalent solutions (the Pareto front). The objectives are often aggregated into a single cost function to reduce the computational cost, though a better approach is to use multiobjective optimization methods to directly identify a set of Pareto-optimal solutions, which can be used by the designer to make more efficient design decisions (instead of making those decisions upfront). Most of the work in multiobjective optimization is focused on MultiObjective Evolutionary Algorithms (MOEAs). While MOEAs are well-suited to handle large, intractable design spaces, they typically require thousands of expensive simulations, which is prohibitively expensive for the problems under study. Therefore, the use of surrogate models in multiobjective optimization, denoted as MultiObjective Surrogate-Based Optimization (MOSBO), may prove to be even more worthwhile than SBO methods to expedite the optimization process. In this paper, the authors propose the Efficient Multiobjective Optimization (EMO) algorithm which uses Kriging models and multiobjective versions of the expected improvement and probability of improvement criterions to identify the Pareto front with a minimal number of expensive simulations. The EMO algorithm is applied on multiple standard benchmark problems and compared against the wellknown NSGA-II and SPEA2 multiobjective optimization methods with promising results.

The use of semantic technology has recently witnessed a huge increase. One of the areas in which this technology is being used increasingly more often is that of context-aware agents. However, the use of ontologies in general and reasoning in particular can rapidly become resource intensive. Certainly if the data set, called the A-Box, used by these agents grows considerably over time. Moreover, in order to create context-aware applications, taking into account a wide range of different data sets and context parameters, agents have to be provided to expose that data. The collaboration between the agents in the system is necessary to correlate the information and augment the intelligence and added value of the context-aware agents. Therefore, there is a need to have a distributed approach by means of a service-platform, where the different agents in a context-aware environment can collaborate. The main focus of this article is on the research on the design of a service-platform for semantic ontology-based context-aware collaboration. The platform architecture to allow the collaboration and scheduling, together with the associated algorithms, will be presented. The engineering and implementation details will be highlighted. By means of detailed UML sequence diagrams, we will present the workflow and collaboration between the different modules in the platform. Additionally, supporting developments, such as the meta-ontology and our ontology generator, OTAGen, will be presented. Furthermore, we will detail how the platform can operate in an autonomous way, taking into account the changing context of the agents in the platform.

Medical time series contain important information about the condition of a patient. However, due to the large amount of data and the staff shortage, it is difficult for physicians to monitor these time series for trends that suggest a relevant clinical detoriation due to a complication or new pathology. This paper proposes a framework that supports physicians in detecting patterns in time series. It has three main tasks. First, the time-dependent data is gathered from heterogeneous sources and the semantics are made explicit by using an ontology. Second, Machine Learning techniques detect trends in the semantic time series data that indicate that a patient has a particular pathology. However, computerized classification techniques are not 100% accurate. Therefore, the third task consists of adding the pathology classification to the ontology with an associated probability and notifying the physician if necessary. The framework was evaluated with an ICU use case, namely detecting sepsis. Sepsis is the number one cause of death in the ICU.

This paper describes a robust and accurate black-box macromodeling technique, in which the constitutive equations combine both closed-form delay operators and low-order rational coefficients. These models describe efficiently electrically long interconnect links. The algorithm is based on an iterative weighted least-squares process and can be interpreted as a generalization of the well-known Vector Fitting. The paper is focused, in particular, on the passivity enforcement of these models. We present two perturbation methods and we show how the accuracy of the models is well-preserved during the passivity enforcement process.

In this paper, we use machine learning technique at the routers to study the link state protocol data to predict the existence of shared risk link groups (SRLG) in the network. In particular, we use the correlation between different link state updates (LSUs) issued by different network nodes (routers) upon failure. The concerned network router then runs a novel Bayesian network based statistical learning process to learn about the possible existence of SRLGs. The decision of this online learning is transferred to the routing information base (RIB) so that it can accordingly modify the routing table for the entire SRLG upon failure detection of one of the candidate node of that particular SRLG and hence reduce the protection switching time.

Many complex, real world phenomena are difficult to study directly using controlled experiments. Instead, the use of computer simulations has become commonplace as a cost effective alternative. However, regardless of Moorepsilas law, performing high fidelity simulations still requires a great investment of time and money. Surrogate modeling (metamodeling) has become indispensable as an alternative solution for relieving this burden. Many surrogate model types exist (support vector machines, Kriging, RBF models, neural networks, ...) but no type is optimal in all circumstances. Nor is there any hard theory available that can help make this choice. The same is true for setting the surrogate model parameters (bias- variance trade-off). Traditionally, the solution to both problems has been a pragmatic one, guided by intuition, prior experience or simply available software packages. In this paper we present a more founded approach to these problems. We describe an adaptive surrogate modeling environment, driven by speciated evolution, to automatically determine the optimal model type and complexity. Its utility and performance is presented on a case study from electronics.

In this paper, the focus is on how context information can be efficiently modeled with an ontology. This ontology can than be used by reasoning algorithms which are based on this context information. This is illustrated with a use case which studies the evolution from a place oriented to a person oriented nurse call system. An ontology was designed which holds the necessary context information. A nurse call algorithm that uses this information was constructed. The CASP Context framework was extended to implement the use case. This framework is bases on an OSGi framework. Rules are formulated to implement the algorithm. OWL was applied to integrate the ontology into the framework. A Web Service interface was designed which allows to insert new information into the Knowledge Base or extract information from it. At last a simulation was set up to show the advantages of the person oriented approach. The results of a performance study are shown as well.

On the one hand, agent-based software platforms are commonly used these days, while on the other hand Semantic Web technologies are also maturing. It is obvious that the combination of these two technologies can bring added value through the creation of Semantic Agent-based frameworks. However, it is also known that these Semantic Web technologies, and the reasoning on ontologies in particular, can rapidly become resource intensive. In order to get a clear view on this problem, we have developed OTAGen, a highly tunable tool to generate customized ontologies and corresponding queries. The generated ontologies can then be used to evaluate at design-time the performance of the Semantic Agent-based platform as a function of the number of ontologies, users and queries.

Simulation and optimization of complex mechanical and electronical systems is a very time consuming and computationally intensive task. Therefore, metamodeling techniques are often used for the efficient exploration of the design space, as they reduce the number of simulations needed. However, constructing such metamodels (or surrogate models) is typically done in a sequential fashion. In this paper we argue that this approach can still be improved. We propose a framework where modeler and simulator interact through a distributed environment, (using established grid computing techniques) thus decreasing model generation and simulation turnaround time.

Frequency-domain rational macromodeling techniques have to be limited to relatively small frequency ranges or to a limited number of poles, mostly due to numerical issues. A pertinent problem in system modeling is therefore to come up with a global broad-band macromodel, given a number of piecewise rational, possibly disjoint, small-band models. We describe an effective implementation procedure, based on orthonormal bandlimited Kautz sequences. As a first step, we show how a truncated Kautz basis can be obtained directly from a judiciously chosen state-space description. Next, having incorporated the bandlimitedness requirement, we obtain imbeddable orthonormal bandlimited Kautz sequences. A numerical procedure for calculating the underlying bandlimited scalar products and Grammians, as applied to piecewise bandlimited state-space data, is implemented and tested

A practical method is developed for selecting the minimal number of lumped elements needed to represent a lossy transmission line if a certain accuracy is desired in a well-defined frequency range. This method, which uses dimensionless transmission line parameters, can be used in a wide range of applications and is also extended to hybrid equivalent circuits, consisting of ideal single lossless lines and resistors. For completeness, a lumped element model for coupled lossy lines is presented which uses the same dimensionless parameters and the same criteria as proposed for single lines. An example of coupled transmission line structure including skin-effect losses illustrates the approach.>